Nearly Tight Bounds for Sandpile Transience on the Grid

TL;DR

This paper establishes nearly tight bounds on the number of grains that can be added to a 2D grid in the Abelian sandpile model before reaching steady state, using electrical network techniques.

Contribution

It provides the first nearly tight bounds for the transience class of the sandpile model on grids, extending to higher dimensions with refined analysis.

Findings

Upper bound of O(n^4 log^4 n] for 2D grids

Lower bound of Ω(n^4) for 2D grids

Bounds extend to d-dimensional grids with similar techniques

Abstract

We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an $O(n^4\log^4{n})$ upper bound and an $\Omega(n^4)$ lower bound for the transience class of the $n \times n$ grid. Our methods naturally extend to $n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper bounds and $\Omega(n^{3d -2})$ lower bounds.

Equations

$$\displaystyle\mathcal{W}\left(S\rightarrow T\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Big{\{}{\mathit{w}}:{\mathit{w}}^{(0)}\in S\text{, }{\mathit{w}}^{(i)}\not\in T\cup\{v_{\textnormal{sink}}\}\text{ for $0\leq i\leq|{\mathit{w}}|-1$}\text{, and }{\mathit{w}}^{(|{\mathit{w}}|)}\in T\Big{\}}$$

$$\displaystyle\mathcal{W}\left(S\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\Bigg{\{}{\mathit{w}}\in\prod_{i=0}^{\infty}V:{\mathit{w}}^{(0)}\in S\text{ and }{\mathit{w}}^{(i)}\neq v_{\textnormal{sink}}\text{ for any $i\geq 0$}\Bigg{\}},$$

$$\underset{\le t}{\min }(w)\stackrel{\mathrm{def}}{=}\underset{0\le \hat{t}\le t}{\min }{w}^{(\hat{t})}$$

$$\underset{\le t}{\max }(w)\stackrel{\mathrm{def}}{=}\underset{0\le \hat{t}\le t}{\max }{w}^{(\hat{t})}.$$

$${\mathbf{\pi }}_u(v)=\frac{1}{\text{deg}(v)}\sum _{x\sim v}{\mathbf{\pi }}_u(x),$$

$$\text{tcl}(G)=O\left(\underset{u,v\in V\setminus \{v_{\text{sink}}\}}{\max }\left(\sum _{x\in V}{\mathbf{\pi }}_u(x)\right){\mathbf{\pi }}_u(v{)}^{-1}\right)$$

$$\text{tcl}(G)=\Omega \left(\underset{u,v\in V\setminus \{v_{\text{sink}}\}}{\max }{\mathbf{\pi }}_u(v{)}^{-1}\right).$$

$${\mathbf{\pi }}_u\left(v\right)=\sum _{w\in \mathcal{W}\left(v\to u\right)}{4}^{-|w|}.$$

$${\mathcal{R}}_{\text{eff}}\left(v_{\text{sink}},u\right){\mathbf{\pi }}_u(v)={\mathcal{R}}_{\text{eff}}\left(v_{\text{sink}},v\right){\mathbf{\pi }}_v(u).$$

$$1/4\le {\mathcal{R}}_{\text{eff}}\left(v_{\text{sink}},u\right)\le 2\log n+1.$$

$${\mathbf{\pi }}_u\left(v\right)\le \left(8\log n+4\right){\mathbf{\pi }}_v\left(u\right).$$

$$\sum _{v\in V}{\mathbf{\pi }}_u\left(v\right)=O\left(u_1{u}_2{\log }^{3}n\right).$$

$${\mathbf{\pi }}_u\left(v\right)=O\left({\mathbf{\pi }}_v\left(u\right)\log n\right),$$

$${\mathbf{\pi }}_v\left(u\right)$$

$$\le {\text{Pr}}_{w\sim {\mathcal{W}}^{{\mathbb{Z}}^2}(u)}[\text{w1 hits v1 before vsink and w2 hits v2 before vsink}]$$

$$=\text{mbox}{Pr}_{w\sim {\mathcal{W}}^{{\mathbb{Z}}^2}(u)}\left[\text{w1 hits v1 before vsink}\right]\cdot \text{mbox}{Pr}_{w\sim {\mathcal{W}}^{{\mathbb{Z}}^2}(u)}\left[\text{w2 hits v2 before vsink}\right]$$

$$={\mathbf{\pi }}_{v_1}^{\text{textsc}{Path}_n}(u_1)\cdot {\mathbf{\pi }}_{v_2}^{\text{textsc}{Path}_n}(u_2).$$

$$\sum _{v\in V}{\mathbf{\pi }}_v\left(u\right)\le \left(\sum _{v_1=1}^n{\mathbf{\pi }}_{v_1}^{\text{textsc}{Path}_n}\left(u_1\right)\right)\left(\sum _{v_2=1}^n{\mathbf{\pi }}_{v_2}^{\text{textsc}{Path}_n}\left(u_2\right)\right).$$

$${\mathbf{\pi }}_{v_1}^{\text{textsc}{Path}_n}\left(u_1\right)=\{\begin{array}{ll}\frac{n+1-u_1}{n+1-v_1}& \text{if v1≤u1},\\ \frac{u_1}{v_1}& \text{if v1>u1}.\end{array}$$

$$\sum _{v_1=1}^n{\mathbf{\pi }}_{v_1}^{\text{textsc}{Path}_n}(u_1)\le u_1+\sum _{v_1=u_1+1}^n\frac{u_1}{v_1}=O(u_1\log n).$$

$${\mathbf{\pi }}_u\left(v\right)\ge \frac{1}{16}{\mathbf{\pi }}_u\left((n,n)\right).$$

$$\text{mbox}{Pr}_{w\sim {\mathcal{W}}^{\text{textsc}Line}(i)}\left[w^{\left(t\right)}=n,\underset{\le t}{\max }(w)=n,\text{and }\underset{\le t}{\min }(w)\ge 1\right]\ge e^{-2-2c}\frac{i}{n^3}.$$

$$\min \left\{\frac{1}{2^n}\sum _{\begin{array}{c}k=\lceil \frac{n}{4}\rceil \\ k\text{ odd}\end{array}}^{\lfloor \frac{3n}{4}\rfloor }\left(\genfrac{}{}{0ex}{}{n}{k}\right),\frac{1}{2^n}\sum _{\begin{array}{c}k=\lceil \frac{n}{4}\rceil \\ k\text{ even}\end{array}}^{\lfloor \frac{3n}{4}\rfloor }\left(\genfrac{}{}{0ex}{}{n}{k}\right)\right\}\ge \frac{2}{5}.$$

$${\mathbf{\pi }}_{(n,n)}\left(u\right)\ge e^{-100}\frac{u_1{u}_2}{n^4}.$$

$${\mathbf{\pi }}_{(n,n)}\left(u\right)\ge \sum _{\begin{array}{c}w\sim \mathcal{W}(u\to (n,n))\\ \text{w1 hits n exactly once}\\ \text{w2 hits n exactly once}\end{array}}{4}^{-|w|}.$$

$$\sum _{t_1,t_2\ge 0}$$

$$\cdot \left(\text{\# of t2-step walks from u2 that stay in [1,n−1] and terminate at n}\right).$$

$$\text{mbox}{Pr}_{w_1\sim \mathcal{W}\left(u_1\right)}\left[w_1^{(t_1)}=n,\underset{\le t_1-1}{\max }(w)=n-1,\underset{\le t_1-1}{\min }(w)\ge 1\right]$$

$$=\frac{\left(\text{\# of t1-step walks from u1 that stay in [1,n−1] and terminate at n}\right)}{2^{t_1}},$$

$${\mathbf{\pi }}_{(n,n)}\left(u\right)\ge \sum _{t_1,t_2\ge 0}$$

$$\cdot \text{mbox}{Pr}_{w_2\sim \mathcal{W}\left(u_2\right)}\left[w_2^{(t_2)}=n,\underset{\le t_2-1}{\max }(w)=n-1,\underset{\le t_2-1}{\min }(w)\ge 1\right].$$

$$\sum _{t_1,t_2\ge 5}$$

$$\cdot \left(\frac{1}{2}\text{mbox}{Pr}_{w_2\sim \mathcal{W}\left(u_2\right)}\left[w_2^{(t_2-1)}=n-1,\underset{\le t_2-1}{\max }(w)=n-1,\underset{\le t_2-1}{\min }(w)\ge 1\right]\right).$$

$${\mathbf{\pi }}_{(n,n)}\left(u\right)$$

Full text

1 Introduction

The Abelian sandpile model is the canonical dynamical system used to study self-organized criticality. In their seminal paper, Bak, Tang, and Wiesenfeld [BTW87] proposed the idea of self-organized criticality to explain several ubiquitous patterns in nature typically viewed as complex phenomena, such as catastrophic events occurring without any triggering mechanism, the fractal behavior of mountain landscapes and coastal lines, and the presence of pink noise in electrical networks and stellar luminosity. Since their discovery, self-organized criticality has been observed in an abundance of disparate scientific fields [Bak96, WPC*+*16], including condensed matter theory [WWAM06], economics [BPR15, SW94], epidemiology [SMM14], evolutionary biology [Phi14], high-energy astrophysics [Asc11, MTN94], materials science [RAM09], neuroscience [BdACA*+*16, LHG07], statistical physics [Dha06, Man91], seismology [SS89], and sociology [KG09]. A stochastic process is a self-organized critical system if it naturally evolves to highly imbalanced critical states where slight local disturbances can completely alter the current state. For example, when pouring grains of sand onto a table, the pile initially grows in a predictable way, but as it becomes steeper and more unstable, dropping a single grain can spontaneously cause an avalanche that affects the entire pile. Self-organized criticality differs from the critical point of a phase transition in statistical physics, because a self-organizing system does not rely on tuning an external parameter. Instead, it is insensitive to all parameters of the model and simply requires time to reach criticality, which is known as the transient period. Natural events empirically operate at a critical point between order and chaos, thus justifying our study of self-organized criticality.

Dhar [Dha90] developed the Abelian sandpile model on finite directed graphs with a sink vertex to further understand self-organized criticality. The Abelian sandpile model, also known as a chip-firing game [BLS91], on a graph with a sink is defined as follows. In each iteration a grain of sand is added to a non-sink vertex of the graph. While any non-sink vertex $v$ contains at least $\text{deg}(v)$ grains of sand, a grain is transferred from $v$ to each of its neighbors. This is known as a toppling. When no vertex can be toppled, the state is stable and the iteration ends. The sink absorbs and destroys grains, and the presence of a sink guarantees that every toppling procedure eventually stabilizes. An important property of the Abelian sandpile model is that the order in which vertices topple does not affect the stable state. Therefore, as the process evolves it produces a sequence of stable states. From the theory of Markov chains, we say that a stable state is recurrent if it can be revisited; otherwise it is transient.

In the self-organized critical state of the Abelian sandpile model on a graph with a sink, transient states have zero probability and recurrent states occur with equal probability [Dha90]. As a result, recurrent configurations model the steady-state behavior of the system. Thus, the natural algorithmic question to ask about self-organized criticality for the Abelian sandpile model is:

Question 1.1**.**

How long in the worst case does it take for the process to reach its steady-state behavior or, equivalently, a recurrent state?

Starting with an empty configuration, if the vertex that receives the grain of sand is chosen uniformly at random in each step, Babai and Gorodezky [BG07] give a simple solution that is polynomial in the number of edges of the graph using a coupon collector argument. In the worst case, however, an adversary can choose where to place the grain of sand in each iteration. Babai and Gorodezky analyze the transience class of the model to understand its worst-case behavior, which is defined as the maximum number of grains that can be added to the empty configuration before the configuration necessarily becomes recurrent. An upper bound for the transience class of a model is an upper bound for the time needed to enter self-organized criticality.

1.1 Results

We give the first nearly tight bounds (up to polylogarithmic factors) for the transience class of the Abelian sandpile model on the $n\times n$ grid with all boundary vertices connected to the sink. This model was first studied in depth by Dhar, Ruelle, Sen, and Verma [DRSV95], and it has since been the most extensively studied Abelian sandpile model due to its role in algebraic graph theory, theoretical computer science, and statistical physics. Babai and Gorodezky [BG07] initially established that the transience class of the grid is polynomially bounded by $O(n^{30})$, which was unexpected because there are graphs akin to the grid with exponential transience classes. Choure and Vishwanathan [CV12] improved the upper bound for the transience class of the grid to $O(n^{7})$ and gave a lower bound of $\Omega(n^{3})$ by viewing the graph as an electrical network and relating the Abelian sandpile model to random walks on the underlying graph. Moreover, they conjectured that the transience class of the grid is $O(n^{4})$, which we answer nearly affirmatively.

Theorem 1.2**.**

The transience class of the Abelian sandpile model on the $n\times n$ grid is $O(n^{4}\log^{4}{n})$.

Theorem 1.3**.**

The transience class of the Abelian sandpile model on the $n\times n$ grid is $\Omega(n^{4})$.

Our results establish how fast the system reaches its steady-state behavior in the adversarial case, and they corroborate empirical observations about natural processes exhibiting self-organized criticality. Our analysis directly generalizes to higher-dimensional cases, giving the following result.

Theorem 1.4**.**

For any integer $d\geq 2$, the transience class of the Abelian sandpile model on the $n^{d}$-sized $d$-dimensional grid is $O(n^{3d-2}\log^{d+2}{n})$ and $\Omega(n^{3d-2})$.

In addition to addressing the main open problem in [BG07] and [CV12], we begin to shed light on Babai and Gorodezky’s inquiry about sequences of graphs that exhibit polynomially bounded transience classes. Specifically, for hypergrids (a family of locally finite graphs with high symmetry) we quantify how the transience class grows as a function of the size and local degree of the graph. When viewed through the lens of graph connectivity, such transience class bounds are surprising because grids have low algebraic connectivity, yet we are able to make global structural arguments using only the fact that grids have low maximum effective resistance when viewed as electrical networks. By doing this, we avoid spectral analysis of the grid and evade the main obstacle in Choure and Vishwanathan’s analysis. Our techniques suggest that low effective resistance captures a different but similar phenomenon to high conductance and high edge expansion for stochastic processes on graphs. This distinction between the role of a graph’s effective resistance and conductance could be an important step forward for building a theory for discrete diffusion processes analogous to the mixing time of Markov chains. We also believe our results have close connections to randomized, distributed optimization algorithms for flow problems [BBD*+*13, BMV12, Meh13, SV16a, SV16b, SV16c], where the dynamics of self-adjusting sandpiles (a Physarum slime mold in their model) are governed by electrical flows and resistances.

1.2 Techniques

Our approach is motivated by the method of Choure and Vishwanathan [CV12] for bounding the transience class of the Abelian sandpile model on graphs using electrical potential theory and the analysis of random walks. Viewing the graph as an electrical network with a voltage source at some vertex and a grounded sink, we give more accurate voltage estimates by carefully considering the geometry of the grid. We use several lines of symmetry to compare escape probabilities of random walks with different initial positions, resulting in a new technique for comparing vertex potentials. These geometric arguments can likely be generalized to other lattice-based graphs. As a result, we get empirically tight inequalities for the sum of all vertex potentials in the grid and the voltage drop between opposite corners of the network.

For many of our voltage bounds, we interpret a vertex potential as an escape probability and decouple the corresponding two-dimensional random walks on the grid into independent one-dimensional random walks on a path graph. Decoupling is the standout technique in this paper, because it allows us to apply classical results about simple symmetric random walks on $\mathbb{Z}$ (such as the reflection principle), which we extend as needed using conditional probability arguments. By reducing from two-dimensional random walks to one-dimensional walks, we utilize standard probabilistic tools including Stirling’s approximation, Chernoff bounds, and the negative binomial distribution. Since we consider many different kinds of events in our analysis, Section 5 is an extensive collection of probability inequalities for symmetric $t$-step random walks on $\mathbb{Z}$ with various boundary conditions. We noticed that some of these inequalities are directly related to problems in enumerative combinatorics without closed-form solutions [ES77].

Lastly, we leverage well-known results about effective resistances of the $n\times n$ grid when viewed as an electrical network. We follow Choure and Vishwanathan in using the potential reciprocity theorem to swap the voltage source with any other non-sink vertex, but we use this theorem repeatedly with the fact that the effective resistance between any non-sink vertex and the sink is bounded between a constant and $O(\log n)$. This approach enables us to analyze tractable one-dimensional random walk problems at the expense of polylogarithmic factors.

2 Preliminaries

2.1 Abelian Sandpile Model

Let $G=(V,E)$ be an undirected multigraph. Throughout this paper all of the graphs we consider have a sink vertex denoted by $v_{\textnormal{sink}}$. The Abelian sandpile model is a dynamical system on a graph $G$ used to study the phenomenon of self-organized criticality. A configuration $\sigma$ on $G$ in the Abelian sandpile model is a vector of nonnegative integers indexed by the non-sink vertices such that $\sigma(v)$ denotes the number of grains of sand on vertex $v$. We say that a configuration is stable if $\sigma(v)<\text{deg}(v)$ for all non-sink vertices and unstable otherwise. An unstable configuration $\sigma$ moves towards stabilization by selecting a vertex $v$ such that $\sigma(v)\geq\text{deg}(v)$ and sending one grain of sand from $v$ to each of its neighboring vertices. This event is called a toppling of $v$, and it creates a new configuration $\sigma^{\prime}$ such that $\sigma^{\prime}(v)=\sigma(v)-\text{deg}(v)$, $\sigma^{\prime}(u)=\sigma(u)+1$ for all vertices $u$ adjacent to $v$, and $\sigma^{\prime}(u)=\sigma(u)$ for all remaining vertices. This procedure eventually reaches a stable state because $G$ has a sink. Moreover, the order in which vertices topple does not affect the final stable state. The initial configuration of the Abelian sandpile model is typically the zero vector, and in each iteration a grain of sand is placed at a vertex (chosen either deterministically or uniformly at random). The system evolves by stabilizing the configuration and then receiving another grain of sand.

A stable configuration $\sigma$ is recurrent if the process can eventually return to $\sigma$. Any state that is not recurrent is transient. Note that once the system enters a recurrent state, it can never visit a transient state. Babai and Gorodezky [BG07] introduced the following notion to upper bound on the number of steps for the Abelian sandpile model to reach self-organized criticality.

Definition 2.1**.**

The transience class of the Abelian sandpile model of $G$ is the maximum number of grains that can be added to the empty configuration before the configuration necessarily becomes recurrent. We denote this quantity by $\textnormal{tcl}(G)$.

In Figure 1 we illustrate the transient configurations in the transient period of the Abelian sandpile model as it advances towards its critical state. We specifically show in this paper that by repeatedly placing grains of sand in the top-left corner of the grid, we maximize the length of the transience period up to a polylogarithmic factor.

In earlier related works, Björner, Lovász, and Shor [BLS91] studied a variant of this process without a sink and characterized the conditions needed for stabilization to terminate. They also related the spectrum of the underlying graph to the rate at which the system converges. In the model we study, an observation by Dhar [Dha90] and Kirchoff’s theorem show that the stable recurrent states of the system are in bijection with the spanning trees of $G$. Choure and Vishwanathan [CV12] show that if every vertex in a configuration has toppled then the configuration is necessarily recurrent, which we use to bound the transience class. The Abelian sandpile model also has broad applications to algorithms and statistical physics, including a direct relation to the $q$-state Potts model and Markov chain Monte Carlo algorithms for sampling random spanning trees [BCFR17, Dha90, JLP15, RS17, Wil10]. For a comprehensive survey on the Abelian sandpile model, see [HLM*+*08].

2.2 Random Walks on Graphs

A walk ${\mathit{w}}$ on $G$ is a sequence of vertices ${\mathit{w}}^{(0)},{\mathit{w}}^{(1)},\dots,{\mathit{w}}^{(t_{\max})}$ such that every ${\mathit{w}}^{(t+1)}$ is a neighbor of ${\mathit{w}}^{(t)}$. We let $t_{\max}=|w|$ denote the length of the walk. A random walk is a process that begins at vertex $w^{(0)}$, and at each time step $t$ transitions from $w^{(t)}$ to $w^{(t+1)}$ such that $w^{(t+1)}$ is chosen uniformly at random from the neighbors of $w^{(t)}$. Note that this definition naturally captures the effect of walking on a multigraph. We consider walks that continue until reaching a set of sink vertices. It will be convenient for our analysis to formally define these following families of walks.

Definition 2.2**.**

For any set of starting vertices $S$ and terminating vertices $T$ in the graph $G$, let

[TABLE]

be the set of finite walks from $S$ to $T$.

Observe that with this definition, walks $w$ of length [math] are permissible if we have $w^{(0)}\in S\cap T$. Throughout the paper it will be convenient to consider random walks from one vertex $u$ to another vertex $v$ or the pair $\{v,v_{\textnormal{sink}}\}$. We denote these cases by the notation $\mathcal{W}\left(u\rightarrow v\right)=\mathcal{W}(\{u\}\rightarrow\{v\}).$ If walks on multiple graphs are being considered, we use $\mathcal{W}^{G}(u\rightarrow v)$ to denote the underlying graph. Lastly, we consider the set of nonterminating walks in our analysis, so it will be useful to define

[TABLE]

which is the set of infinite walks from $S$. An analogous definition follows when $S=\{u\}$.

The focus of our study is the $n\times n$ grid graph, denoted by $\textsc{Square}_{n}$. Similar to previous works, we do not follow the usual graph-theoretic convention of using $n$ to denote vertex count. We formally define the one-dimensional projection of $\textsc{Square}_{n}$ to be $\textsc{Path}_{n}$, which has the vertex set $\{1,2,\dots,n\}\cup\{v_{\textnormal{sink}}\}$ and edges between $i$ and $i+1$ for every $1\leq i\leq n-1$, as well as two edges connecting $v_{\textnormal{sink}}$ to $1$ and $n$. Thus, $v_{\textnormal{sink}}$ can be viewed as [math] and $n+1$. If we remove the sink (which can be thought of as letting $v_{\textnormal{sink}}=\pm\infty$) then the resulting graph is the one-dimensional line with vertices $i\in\mathbb{Z}$ and edges between every pair $(i,i+1)$. We denote this graph by Line and use the indices $i$, $j$, and $k$ to represent its vertices. Analyzing random walks on Line is critical to our analysis, and it will be useful to record the minimum and maximum position of $t$-step walks.

Definition 2.3**.**

For an initial position $i\in\mathbb{Z}$ and walk ${\mathit{w}}\in\mathcal{W}(i)$ on Line, let the $t$-step minimum and maximum positions be

[TABLE]

and

[TABLE]

We construct $\textsc{Square}_{n}$ similarly. Its vertices are $\left\{1,2,\ldots,n\right\}\times\left\{1,2,\ldots,n\right\}\cup\{v_{\textnormal{sink}}\}$, and its edges connect any pair of vertices that differ in one coordinate. Vertices on the boundary have edges connected to $v_{\textnormal{sink}}$ so that every non-sink vertex has degree $4$. With this definition of $\textsc{Square}_{n}$, each corner vertex has two edges to $v_{\textnormal{sink}}$ and non-corner vertices on the boundary share one edge with $v_{\textnormal{sink}}$. Since all vertices correspond to pairs of coordinates, we use the vector notation $\boldsymbol{\mathit{u}}=(\boldsymbol{\mathit{u}}_{1},\boldsymbol{\mathit{u}}_{2})$ to denote coordinates on the grid, as it easily extends to higher dimensions. Throughout the paper, boldfaced variables denote vectors. A $t$-step random walk on $\textsc{Square}_{n}$ naturally induces a $(t_{\max}+1)\times 2$ matrix. We can decouple such a walk ${\mathit{w}}$ into its horizontal and vertical components, using the notation ${\mathit{w}}_{1}$ for the change in position of the first coordinate and ${\mathit{w}}_{2}$ for the change in position of the second coordinate. In general we use the notation ${\mathit{w}}_{\widehat{d}}$ to index into one of the dimensions $1\leq\widehat{d}\leq d$ of a $d$-dimensional walk. We do not record duplicate positions when the walk takes a step in a dimension different than $\widehat{d}$, so we have $\left|{\mathit{w}}\right|=\left|{\mathit{w}}_{1}\right|+\left|{\mathit{w}}_{2}\right|-1$ when $d=2$ since the initial vertex is present in both ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$.

2.3 Electrical Networks

Vertex potentials are central to our analysis. They have close connections with electrical voltages and belong to the class of harmonic functions [DS84]. We analyze their relation to the transience class of general graphs. For any non-sink vertex $u$, we can define a unique potential vector $\boldsymbol{\pi}_{u}$ such that $\boldsymbol{\pi}_{u}(u)=1$, $\boldsymbol{\pi}_{u}(v_{\textnormal{sink}})=0$, and for all other vertices $v\in V\setminus\{u,v_{\textnormal{sink}}\}$ we have

[TABLE]

where the sum is over the neighbors of $v$. Thus, $\boldsymbol{\pi}_{u}(v)$ denotes the potential at $v$ when the boundary conditions are set to $1$ at $u$ and [math] at the sink. Since we analyze potential vectors in both $\textsc{Path}_{n}$ and $\textsc{Square}_{n}$, we use superscripts to denote the graph when context is unclear.

Choure and Vishwanathan showed that we can give upper and lower bounds on the transience class using potentials, which we rephrase in the following theorem.

Theorem 2.4** ([CV12]).**

If $G$ is a graph such that the degree of every non-sink vertex is bounded by a constant,

[TABLE]

and

[TABLE]

All non-sink vertices have degree $4$, so we can apply Theorem 2.4 to $\textsc{Square}_{n}$.

The following combinatorial interpretations of potentials as random walks is fundamental to our investigation of the transience class of $\textsc{Square}_{n}$. Note that we use boldfaced vector variables for non-sink vertices in $\textsc{Square}_{n}$ as they can be identified by their coordinates.

Fact 2.5** ([DS84]).**

For any graph $G$ and non-sink vertex $u$, the potential $\boldsymbol{\pi}_{u}(v)$ is the probability of a random walk starting at $v$ and reaching $u$ before $v_{\textnormal{sink}}$.

Lemma 2.6**.**

Let $\boldsymbol{\mathit{u}}$ be a non-sink vertex of $\textsc{Square}_{n}$. For any vertex $\boldsymbol{\mathit{v}}$, we have

[TABLE]

We defer the proof of Lemma 2.6 to Appendix A.

A systematic treatment of the connection between random walks and electrical networks can be found in the monograph by Doyle and Snell [DS84] or the survey by Lovász [Lov93]. The following lemma is a key result for our investigation, which states that a voltage source and a measurement point can be swapped at the expense of a distortion in the potential equal to the ratio of the effective resistances between the sink and the two vertices. The effective resistance between a pair of vertices $u$ and $v$, denoted as $\mathcal{R}_{\textnormal{eff}}(u,v)$, can be formalized in several ways. In the electrical interpretation [DS84], effective resistance can be viewed as the voltage needed to send one unit of current from $u$ to $v$ if every edge in $G$ is a unit resistor. For a linear algebraic definition of effective resistance see [ESVM*+*11].

Lemma 2.7** ([CV12, Potential Reciprocity]).**

Let $G$ be a graph (not necessarily degree-bounded) with sink $v_{\textnormal{sink}}$. For any pair of vertices $u$ and $v$, we have

[TABLE]

This statement is particularly powerful for $\textsc{Square}_{n}$, because the effective resistance between any pair of vertices is bounded between a constant and $O(\log n)$. The following lemma makes use of a classical result that can be obtained using Thompson’s principle of the electrical flow [DS84].

Lemma 2.8**.**

For any non-sink vertex $\boldsymbol{\mathit{u}}$ in $\textsc{Square}_{n}$,

[TABLE]

We give the proof of Lemma 2.8 in Appendix A. When used together, Lemma 2.7 and Lemma 2.8 imply the following result, which allows us to conveniently swap the source vertex when computing potentials.

Lemma 2.9**.**

For any non-sink vertices $\boldsymbol{\mathit{u}}$ and $\boldsymbol{\mathit{v}}$ in $\textsc{Square}_{n}$, we have

[TABLE]

Voltages and flows on electrical networks are central to many recent developments in algorithmic graph theory (e.g. modern maximum flow algorithms and interior point methods [CKM*+*11, Mad13]). The convergence of many of these algorithms depend on the extremal voltage values of the electrical flow that they construct. As a result, we believe some of our techniques are relevant to the grid-based instantiations of these algorithms.

3 Upper Bounding the Transience Class

In this section we prove the upper bound in Theorem 1.2 for the transience class of the Abelian sandpile model on the square grid. Our proof follows the framework of Choure and Vishwanathan in that we use Theorem 2.4 to reduce the proof to bounding the following two quantities for any non-sink vertex $\boldsymbol{\mathit{u}}\in V(\textsc{Square}_{n})$:

• •

We upper bound the potential sum $\sum_{\boldsymbol{\mathit{v}}\in V}\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{v}})$.

• •

We lower bound the potential $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{v}})$ for all non-sink vertices $\boldsymbol{\mathit{v}}$.

By symmetry we assume without loss of generality that $\boldsymbol{\mathit{u}}$ is in the top-left quadrant of $\textsc{Square}_{n}$ (i.e., we have $1\leq\boldsymbol{\mathit{u}}_{1},\boldsymbol{\mathit{u}}_{2}\leq\lceil n/2\rceil$). The principal idea is to use reciprocity from Lemma 2.7 and effective resistance bounds from Lemma 2.8 to swap source vertices and bound $\boldsymbol{\pi}_{\boldsymbol{\mathit{v}}}(\boldsymbol{\mathit{u}})$ instead, at the expense of a $O(\log{n})$ factor. The second key idea is to interpret potentials as random walks using Fact 2.5 and then decouple two-dimensional walks on $\textsc{Square}_{n}$ into separate horizontal and vertical one-dimensional walks on $\textsc{Path}_{n}$. Using well-studied properties of one-dimensional random walks, we achieve nearly tight bounds on $\textnormal{tcl}(\textsc{Square}_{n})$.

We note that there is a natural trade-off in the choice of the source vertex $\boldsymbol{\mathit{u}}$. Setting $\boldsymbol{\mathit{u}}$ near the boundary decreases vertex potentials because a random walk has a higher probability of escaping to $v_{\textnormal{sink}}$ instead of $\boldsymbol{\mathit{u}}$. This improves the upper bound of the sum of vertex potentials, but it weakens the lower bound of the minimum vertex potential. For vertices $\boldsymbol{\mathit{u}}$ that are not near the boundary, the opposite is true. Therefore, we account for the choice of $\boldsymbol{\mathit{u}}$ in our bounds.

3.1 Upper Bounding the Potential Sum

Lemma 3.1**.**

For any non-sink vertex $\boldsymbol{\mathit{u}}$ in $\textsc{Square}_{n}$, we have

[TABLE]

Proof.

We use Fact 2.5 and Lemma 2.6 to interpret vertex potentials as random walks. We can omit $v_{\textnormal{sink}}$ because any random walk starting there immediately terminates. By Lemma 2.9,

[TABLE]

so we apply the random walk interpretation to potentials starting at $\boldsymbol{\mathit{u}}$ instead of $\boldsymbol{\mathit{v}}$. Consider one such walk ${\mathit{w}}\in\mathcal{W}(\boldsymbol{\mathit{u}}\rightarrow\boldsymbol{\mathit{v}})$ and its one-dimensional decompositions ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$. The probability of a walk from $\boldsymbol{\mathit{u}}$ reaching $\boldsymbol{\mathit{v}}$ is equal to the probability that two interleaved walks in $\textsc{Path}_{n}$ starting at $\boldsymbol{\mathit{u}}_{1}$ and $\boldsymbol{\mathit{u}}_{2}$ are present on $\boldsymbol{\mathit{v}}_{1}$ and $\boldsymbol{\mathit{v}}_{2}$, respectively, at the same time before either hits their one-dimensional sink $v_{\textnormal{sink}}=\{0,n+1\}$.

If we remove the restriction that these walks are present on $\boldsymbol{\mathit{v}}_{1}$ and $\boldsymbol{\mathit{v}}_{2}$ at the same time and only require that they visit $\boldsymbol{\mathit{v}}_{1}$ and $\boldsymbol{\mathit{v}}_{2}$ before hitting $v_{\textnormal{sink}}$, then each of these less restricted walks ${\mathit{w}}_{d}$ belongs to the class $\mathcal{W}^{\textsc{Path}_{n}}\left(\boldsymbol{\mathit{u}}_{d}\rightarrow\boldsymbol{\mathit{v}}_{d}\right).$ Viewing a walk $w$ on $\textsc{Square}_{n}$ as infinite walk on the lattice $\mathbb{Z}^{2}$ induces independence between $w_{1}$ and $w_{2}$. Thus, we obtain the upper bound

[TABLE]

Summing over all choices of $\boldsymbol{\mathit{v}}=(\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})$ gives

[TABLE]

The potentials of vertices in $\textsc{Path}_{n}$ have the following closed-form solution, as shown in [DS84]:

[TABLE]

Splitting the sum at $\boldsymbol{\mathit{u}}_{1}$ and using the fact that potentials are escape probabilities, we have

[TABLE]

We similarly obtain an upper bound of $O(\boldsymbol{\mathit{u}}_{2}\log n)$ in the other dimension. These bounds along with the initial $O(\log n)$ overhead from swapping $\boldsymbol{\mathit{u}}$ and $\boldsymbol{\mathit{v}}$ gives the desired upper bound. ∎

3.2 Lower Bounding the Minimum Potential

The more involved part of this paper proves a lower bound for the minimum vertex potential $\min_{\boldsymbol{\mathit{v}}\in V\setminus\{v_{\textnormal{sink}}\}}\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{v}})$ as a function of a fixed vertex $\boldsymbol{\mathit{u}}=(\boldsymbol{\mathit{u}}_{1},\boldsymbol{\mathit{u}}_{2})$. Recall that we assumed without loss of generality that $\boldsymbol{\mathit{u}}$ is in the top-left quadrant of $\textsc{Square}_{n}$. We first prove that the minimum potential occurs at vertex $(n,n)$, the corner farthest from $\boldsymbol{\mathit{u}}$. Using Lemma 2.9 to swap $\boldsymbol{\mathit{u}}$ and $(n,n)$ at the expense of a $\Omega(1/\log{n})$ factor, we reduce the problem to giving a lower bound for $\boldsymbol{\pi}_{(n,n)}(\boldsymbol{\mathit{u}})$. Then we decompose walks ${\mathit{w}}\in\mathcal{W}(\boldsymbol{\mathit{u}}\rightarrow\{(n,n),v_{\textnormal{sink}}\})$ into their one-dimensional walks ${{\mathit{w}}_{1}\in\mathcal{W}^{\textsc{Path}_{n}}(\boldsymbol{\mathit{u}}_{1})}$ and ${\mathit{w}}_{2}\in\mathcal{W}^{\textsc{Path}_{n}}(\boldsymbol{\mathit{u}}_{2})$, and we interpret $\boldsymbol{\pi}_{(n,n)}(\boldsymbol{\mathit{u}})$ as the probability that the individual processes ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$ are present on $n$ at the same time before either walk leaves the interval $[1,n]$. Walks on Line that meet at $n$ before leaving the interval $[1,n]$ are equivalent to walks on $\textsc{Path}_{n}$ that meet at $n$ before terminating at $v_{\textnormal{sink}}$. Lastly, we use conditional probabilities to analyze walks on Line instead of walks on $\textsc{Path}_{n}$ in order to leverage well-known facts about simple symmetric random walks.

To lower bound the desired probability $\boldsymbol{\pi}_{(n,n)}(\boldsymbol{\mathit{u}})$, we show that a subset of $\mathcal{W}(\boldsymbol{\mathit{u}}\rightarrow(n,n))$ of interleaved one-dimensional walks starting at $\boldsymbol{\mathit{u}}_{1}$ and $\boldsymbol{\mathit{u}}_{2}$ that first reach $n$ in approximately the same number of steps has a sufficient amount of probability mass. We prove this by observing that the distributions of the number of steps for the walks to first reach $n$ without leaving the interval $[1,n]$ are concentrated around $(n-\boldsymbol{\mathit{u}}_{1})^{2}$ and $(n-\boldsymbol{\mathit{u}}_{2})^{2}$, respectively. Consequently, we show that this distribution is approximately uniform in an $\Theta(n^{2})$ length interval, with each $t$-step having probability $\Omega(\boldsymbol{\mathit{u}}_{1}/n^{3})$ and $\Omega(\boldsymbol{\mathit{u}}_{2}/n^{3})$. We then use Chernoff bounds to show that both walks take approximately the same number of steps with constant probability. Combining these facts, we give the desired lower bound $\Omega(\boldsymbol{\mathit{u}}_{1}\boldsymbol{\mathit{u}}_{2}/n^{4})$.

3.2.1 Opposite Corner Minimizes Potential

We first show that the corner vertex $(n,n)$ has the minimum potential up to a constant factor. Viewing potentials as escape probabilities, we utilize the geometry of the grid to construct maps between sets of random walks that prove the potential of an interior vertex is greater than its axis-aligned projection to the boundary of the grid. We defer the proof of Lemma 3.2 to Appendix B.

Lemma 3.2**.**

If $\boldsymbol{\mathit{u}}$ is a vertex in the top-left quadrant of $\textsc{Square}_{n}$, then for any non-sink vertex $\boldsymbol{\mathit{v}}$ we have

[TABLE]

3.2.2 Lower Bounding Corner Potential

By decomposing two-dimensional walks on $\textsc{Square}_{n}$ that start at $\boldsymbol{\mathit{u}}$ into one-dimensional walks on Line, our lower bound relies on showing that there is a $\Theta(n^{2})$ length interval such that each one-dimensional walk of a fixed length in this interval has probability $\Omega(\boldsymbol{\mathit{u}}_{1}/n^{3})$ or $\Omega(\boldsymbol{\mathit{u}}_{2}/n^{3})$, respectively, of remaining above [math] and reaching $n$ for the first time upon termination. For our purposes, lower bounds for this probability will suffice, and they follow from the following key property for one-dimensional walks that we prove in Section 5.

Lemma 3.3**.**

*Let $n\in\mathbb{Z}_{\geq 1}$ and $1\leq i\leq\lceil n/2\rceil$ be any starting position. For any constant $c>4$ and any $t\in\mathbb{Z}$ such that $n^{2}/c\leq t\leq n^{2}/4$ with $t\equiv n-i\pmod{2}$, a simple symmetric random walk ${\mathit{w}}$ on $\mathbb{Z}$ satisfies *

[TABLE]

Using Lemma 3.3 with the following lemma, we give a lower bound for $\boldsymbol{\pi}_{(n,n)}(\boldsymbol{\mathit{u}})$, the probability that a walk starting from $\boldsymbol{\mathit{u}}$ reaches $(n,n)$ before $v_{\textnormal{sink}}$. Lemma 3.4 is a consequence of a Chernoff bound, and we defer its proof to Appendix B.

Lemma 3.4**.**

For all $n\geq 10$, we have

[TABLE]

Lemma 3.5**.**

For all $n\geq 10$ and any vertex $\boldsymbol{\mathit{u}}$ in the top-left quadrant of $\textsc{Square}_{n}$, we have

[TABLE]

Proof.

We decouple each walk ${\mathit{w}}\in\mathcal{W}(\boldsymbol{\mathit{u}}\rightarrow(n,n))$ into its horizontal walk ${\mathit{w}}_{1}\in\mathcal{W}^{\textsc{Line}}(\boldsymbol{\mathit{u}}_{1})$ and vertical walk ${\mathit{w}}_{2}\in\mathcal{W}^{\textsc{Line}}(\boldsymbol{\mathit{u}}_{2})$. The potential $\boldsymbol{\pi}_{(n,n)}\left(\boldsymbol{\mathit{u}}\right)$ can be interpreted as the probability that ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$ visit $n$ at the same time before either leaves the interval $[1,n]$. We can further decompose $t$-step walks on $\textsc{Square}_{n}$ into those that take $t_{1}$ steps in the horizontal direction and $t_{2}$ in the vertical direction. Considering restricted instances where ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$ visit $n$ exactly once, we obtain the following bound by Lemma 2.6:

[TABLE]

Accounting for all the ways that two one-dimensional walks can be interleaved, the right hand side of Equation 1 is

[TABLE]

Observing that

[TABLE]

it follows from Equation 1 that

[TABLE]

By our choice of $n$ and $\boldsymbol{\mathit{u}}$, the right hand side of inequality above equals

[TABLE]

Letting $t=t_{1}+t_{2}$, we further refine the set of two-dimensional walks so that $t\in[\frac{1}{5}n^{2},\frac{1}{4}n^{2}]$ and $t_{1},t_{2}\in[\frac{1}{4}t,\frac{3}{4}t]$ while capturing a sufficient amount of probability mass for a useful lower bound. Note that the parities of $t_{1}$ and $t_{2}$ satisfy $t_{1}\equiv n-\boldsymbol{\mathit{u}}_{1}\pmod{2}$ and $t_{2}\equiv n-\boldsymbol{\mathit{u}}_{2}\pmod{2}$ for valid walks. Let $I$ be an indexing of all such pairs $(t_{1},t_{2})$. Working from Equation $2$, we have

[TABLE]

For the first inequality, we can apply Lemma 3.3 because

[TABLE]

For the second inequality, we group pairs $(t_{1},t_{2})$ by their sum $t=t_{1}+t_{2}$ and apply Lemma 3.4. The number of $t\in[\frac{1}{5}n^{2},\frac{1}{4}n^{2}]$ with either parity restriction is at least $\lfloor\frac{1}{40}n^{2}\rfloor\geq\frac{1}{50}n^{2}$. ∎

3.3 Proof of Theorem 1.2

We now combine the upper bound for the sum of potentials given by Lemma 3.1 and the lower bounds in Section 3.2 to obtain the overall upper bound for the transience class of the grid.

Proof.

For any $\boldsymbol{\mathit{u}}=(\boldsymbol{\mathit{u}}_{1},\boldsymbol{\mathit{u}}_{2})$ in the top-left quadrant of $\textsc{Square}_{n}$, we have

[TABLE]

The first inequality follows from Lemma 3.2, the second from Lemma 2.9, and the third from Lemma 3.5 and Lemma 3.1. The result follows from Theorem 2.4. ∎

4 Lower Bounding the Transience Class

In this section we lower bound $\textnormal{tcl}(\textsc{Square}_{n})$ using techniques similar to those in Section 3. Since the lower bound in Theorem 2.4 considers the maximum inverse vertex potential over all pairs of non-sink vertices $u$ and $v$, it suffices to upper bound $\pi_{(n,n)}((1,1))$. We lower bound vertex potentials by decomposing two-dimensional walks on $\textsc{Square}_{n}$ into one-dimensional walks on Line and then upper bound the probability that a $t$-step walk on Line starting at $1$ and ending at $n$ does not leave the interval $[1,n]$. More specifically, our upper bound for $\pi_{(n,n)}((1,1))$ follows from Lemma 4.1 (which we prove in Section 5) and Fact 4.2.

Lemma 4.1**.**

For all $n\geq 20$ and $t\geq n-1$, we have

[TABLE]

Fact 4.2**.**

For any nonnegative integer $t_{1}$, we have

[TABLE]

Proof.

This follows directly from the negative binomial distribution. Observe that

[TABLE]

as desired. ∎

By decoupling the two-dimensional walks in a way similar to the proof of Lemma 3.5, we apply Lemma 4.1 to the resulting one-dimensional walks to achieve the desired upper bound.

Lemma 4.3**.**

For all $n\geq 20$, we have

[TABLE]

Proof.

Analogous to our lower bound for $\pi_{(n,n)}((1,1))$, decouple each walk ${{\mathit{w}}\in\mathcal{W}((1,1)\rightarrow(n,n))}$ into its horizontal walk ${\mathit{w}}_{1}\in\mathcal{W}^{\textsc{Line}}(1)$ and its vertical walk ${\mathit{w}}_{2}\in\mathcal{W}^{\textsc{Line}}(1)$. We view $\boldsymbol{\pi}_{(n,n)}\left((1,1)\right)$ as the probability that the walks ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$ are present on $n$ at the same time before either leaves the interval $[1,n]$. Letting $t_{1}$ be the length of $w_{1}$ and $t_{2}$ be the length of $w_{2}$, we relax the conditions on the one-dimensional walks and only require that ${\mathit{w}}_{1}$ and ${\mathit{w}}_{2}$ both are present on $n$ at the final step $t=t_{1}+t_{2}$. Note that now both walks could have previously been present on $n$ at the same time before terminating. This gives the upper bound

[TABLE]

Nesting the summations gives

[TABLE]

Using Fact 4.2, we can upper bound the inner sum by

[TABLE]

Factoring out this term from the initial expression completes the proof. ∎

The upper bound on the maximum term in the right hand side of Lemma 4.3 follows immediately from Lemma 4.1. Now we upper bound the summation in the right hand side of Lemma 4.3 using a simple Lemma 4.1.

Lemma 4.4**.**

If $n\geq 20$ and ${\mathit{w}}\sim\mathcal{W}^{\textsc{Line}}(1)$, we have

[TABLE]

Proof.

We first split the sum into

[TABLE]

We will bound both terms by $O(1/n)$. The upper bound for the first term follows immediately from Lemma 4.1 and the fact that we are summing $n^{2}+1$ terms:

[TABLE]

To upper bound the second summation, we again use Lemma 4.1. When $t>n^{2}$, we have

[TABLE]

Since $64(n/t)^{3}$ is a decreasing function in $t$,

[TABLE]

Therefore, we can bound the infinite sum by the integral

[TABLE]

which concludes the proof since we bounded both halves of the sum by $O(1/n)$. ∎

4.1 Proof of Theorem 1.3

We can now easily combine the lemmas in this section with the bounds that relate vertex potentials to the lower bound for the transience class of $\textsc{Square}_{n}$.

Proof.

Applying Lemma 4.3 and then Lemma 4.1 and Lemma 4.4, it follows that

[TABLE]

Therefore, $\pi_{(n,n)}((1,1))^{-1}=\Omega(n^{4})$. By Theorem 2.4 it follows that $\textnormal{tcl}(\textsc{Square}_{n})=\Omega(n^{4})$. ∎

5 Simple Symmetric Random Walks

Our proofs for upper and lower bounding the sandpile transience class on the grid heavily relied on decoupling two-dimensional walks into two independent one-dimensional walks since they are easier to analyze. This claim is immediately apparent when working with vertex potentials for one-dimensional walks on the path, which we used in the proof of Lemma 3.1.

However, we assumed two essential lemmas about one-dimensional walks to prove the lower and upper bound of the minimum vertex potential. Consequently, in this section we examine the probability

[TABLE]

and we prove these necessary lower and upper bounds in Lemma 3.3 and Lemma 4.1 by extending previously known properties of simple symmetric random walks on $\mathbb{Z}$. The key ideas in these proofs are that: the position of a walk in one dimension follows the binomial distribution; the number of walks reaching a maximum position in a fixed number of steps has an explicit formula; and there are tight bounds for binomial coefficients via Stirling’s approximation.

The properties we need do not immediately follow from previously known facts because we require conditions on both the minimum and maximum positions. Section 5.3 gives proofs of the known explicit expressions for the maximum and minimum position of a walk, along with several other useful facts that follow from this proof. In Section 5.4 we apply Stirling’s bound to give accurate lower bounds on a range of binomial coefficients. Sections 5.5 and 5.6 prove several necessary preliminary lower bound lemmas. We prove Lemma 3.3 at the end of Section 5.6. In Section 5.7 we give necessary upper bound lemmas and a proof of Lemma 4.1.

5.1 Lower Bounding Equation 2

To lower bound Equation 2, we split the desired probability into the product of two probabilities using the definition of conditional probability. Then we prove lower bounds for each.

• •

In Lemma 5.6 we show for $t\in\Theta(n^{2})$, the probability that a walk on $\mathbb{Z}$ starting at ${1\leq i\leq\lceil n/2\rceil}$ is $\mbox{{Pr}}_{{\mathit{w}}\sim\mathcal{W}(i)}\left[{{\min_{\leq t}({{\mathit{w}}})}\geq 1}\right]=\Omega\left(i/n\right).$

• •

In Lemma 5.8 and Lemma 5.7 we bound the probability that a walk starting at $1\leq i\leq\lceil n/2\rceil$ of length $t\in\Theta(n^{2})$ reaches $n$ at step $t$ without going above $n$, conditioned on never dropping below $1$:

[TABLE]

Lemma 3.3 immediately follows multiplying these two bounds together. This division allows us to separate proving a minimum and maximum, and in turn simplifies applying known bounds on binomial distributions. Specifically, Lemma 5.6 is an immediate consequence of explicit expressions for the minimum point of a walk and bounds on binomial coefficients, both of which will be given rigorous treatment in Section 5.3.

These proofs will also output a known explicit expression for the probability of the walk reaching $n$ at step $t$, while only staying to its left. All that remains then is to condition the walk to not go to the left of $1$. Note that $1$ is in the opposite direction of $n$, with respect to the starting position $i$. We formally show that the probability of reaching $n$ without going above $n$ only improves if the walk cannot move too far in the wrong direction, but only for $t\leq(n-i+1)^{2}$, thus giving the reason we need to upper bound $t$ by $n^{2}/4$.

5.2 Upper Bounding Equation 2

The desired lemma only concerns walks starting at $i=1$, which will be critical for our proof. The key idea will then be to split the walk in half and consider the probability that the necessary conditions are satisfied in the first $t/2$ steps and in the second $t/2$ steps. The midpoint of the walk at $t/2$ steps can be any point in the interval $[1,n]$, so we must sum over all these possible midpoints. Removing the upper and lower bound conditions, respectively, will then give the upper bound in Lemma 5.9:

[TABLE]

Due to the first $t/2$ step walk starting at $1$ and the second $t/2$ step walk ending at $n$, the conditions ${\min_{\leq t}({{\mathit{w}}})}\geq 1$ for the first walk and ${\max_{\leq\lceil\frac{t}{2}\rceil}({{\mathit{w}}})}=n$ for the second walk will be the difficult property for each walk to satisfy, respectively. Next we apply facts proved in Section 5.3 to obtain expressions for each term within the summation. The remainder of the upper bound analysis will then focus on bounding those expressions.

5.3 Maximum Position of a Walk

As previously mentioned, our proofs mostly leverage well-known facts about the maximum/minimum position of a random walk along with corresponding bounds for these probabilities. This section will first give the result regarding maximum/minimum position of walks and a connection to Stirling’s approximation.

Observe that if we are only concerned with a single end point, we can fix the starting location at [math] by shifting accordingly. In these cases, the following bounds are well known in combinatorics.

Fact 5.1** ([RB79]).**

For any $t,n\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Proof.

For any $k\leq n$, consider a walk ${\mathit{w}}\in\mathcal{W}\left(0\right)$ that satisfies $w^{(t)}=k$ and ${\max_{\leq t}({{\mathit{w}}})}\geq n$. Let $t^{*}$ be the first time that $w^{(t^{*})}=n$, and construct the walk $m$ ending at $2n-k$ such that

[TABLE]

This reflection map is a bijection, so for $k\leq n$ we have

[TABLE]

Subtracting the probability of the maximum position being at least $n+1$ gives

[TABLE]

Summing over all $k\leq n$, we have

[TABLE]

Considering the parity of $t$ and $n$ completes the proof. ∎

The proof above contains two intermediate expressions for probabilities similar to the ones we want to bound.

Fact 5.2**.**

For any integers $n\geq 0$ and $k\leq n$, we have

[TABLE]

Fact 5.3**.**

Let $t,n\in\mathbb{Z}_{\geq 0}$. For any integer $k\leq n$,

[TABLE]

Proof.

Using Fact 5.1 and analyzing the parity of the walks gives

[TABLE]

as desired. ∎

5.4 Lower Bounding Binomial Coefficients

Ultimately, our goal is to give strong lower bounds on closely related probabilities to the ones above. To do so, we need to use various bounds on binomial coefficients that are consequences of Stirling’s approximation.

Fact 5.4** (Stirling’s Approximation).**

For any positive integer $n$, we have

[TABLE]

An immediate consequence of this is a concentration bound on binomial coefficients.

Fact 5.5**.**

Let $c,n\in\mathbb{R}_{>0}$ such that $c\sqrt{n}<n$. For any $k\in[(n-c\sqrt{n})/2,(n+c\sqrt{n})/2]$, we have

[TABLE]

Proof.

We directly substitute Stirling’s approximation to the definition of binomial coefficients to get

[TABLE]

as desired. ∎

5.5 Lower Bounding the Minimum Position

We now bound the probability of the minimum position of a walk in $\mathcal{W}(i)$ being at least $1$ after $t$ steps.

Lemma 5.6**.**

For any positive integer $n$, initial position $1\leq i\leq\lceil n/2\rceil$, and constant $c>4$, if we have $t\in[n^{2}/c,n^{2}/4]$, then

[TABLE]

Proof.

First observe that

[TABLE]

By symmetry, this sum is

[TABLE]

For each $0\leq k\leq i-1$, Fact 5.1 implies that

[TABLE]

By assumption $k\leq k+1\leq i\leq n\leq\sqrt{ct}$, so applying Fact 5.5 gives

[TABLE]

because $t\leq n^{2}/4$. Summing over $0\leq k\leq i-1$ gives the desired bound. ∎

5.6 Lower Bounding the Final and Maximum Position

Similarly, we can use binomial coefficient approximations to bound the probability of a $t$-step walk terminating at $n$ while never moving to a position greater than $n$.

Lemma 5.7**.**

For any initial position $1\leq i\leq\lceil n/2\rceil$ and any $\max\{n,n^{2}/c\}\leq t\leq n^{2}/4$ with $t\equiv n-i\pmod{2}$, we have

[TABLE]

Proof.

By symmetry we rewrite the probability as

[TABLE]

Fact 5.3 gives that this probability equals to

[TABLE]

We can separately bound the last two terms according to the assumptions on $t$ and $i$. Setting $i=0$ minimizes $\binom{t}{(t+n-i)/2}$ for all $i\geq 0$. Setting $i=\lceil n/2\rceil$ in the numerator, $i=0$ in the denominator, and $t=n^{2}/4$ minimizes $(2(n-i+1))/(t+n+2)$. It follows that

[TABLE]

We reapply Fact 5.5 with the observation that $n\leq\sqrt{ct}$ to get

[TABLE]

as desired. ∎

It remains to condition upon the minimum of a walk. This hinges upon the following statement about moving in the wrong direction only decreasing the probability a walk starting at some $1\leq i\leq\lceil n/2\rceil$ ending at $n$ without ever going past $n$.

Lemma 5.8**.**

For any $1\leq i\leq\lceil n/2\rceil$, at any step $t\leq n^{2}/4$ with $t\equiv n-i\pmod{2}$, we have

[TABLE]

Proof.

Condition on ${\min_{\leq t}({{\mathit{w}}})}<1$ and consider the first time $\widehat{t}$ the walk hits [math]. This means $i\equiv\widehat{t}\pmod{2}$ and in turn $n\equiv t-\widehat{t}\pmod{2}$. The probability of ${\max_{\leq t}({{\mathit{w}}})}={\mathit{w}}^{(t)}=n$ via the walk in steps $\widehat{t}+1,\dots,t$ is then at most

[TABLE]

Note that we have inequality since it is possible that we already have ${\max_{\leq\widehat{t}}({{\mathit{w}}})}>n$. Therefore, it suffices to show for any $n$ and any $1\leq\widehat{t}\leq t$ we have

[TABLE]

There are two variables that are shifted from one side of the inequality to the other, the starting position of the walk and the number of steps. In order to prove the inequality, we will show that both taking more steps and starting further to the right will only improve the probability of ending at $n$ and not going above $n$.

We begin by showing that taking more steps will only improve this probability:

[TABLE]

There is no guarantee that $t\equiv n\pmod{2}$, so we consider $t$ or $t-1$ steps depending on parity. We are guaranteed that $t-1\geq t-\widehat{t}$ since $\widehat{t}\geq 1$, so without loss of generality, we assume $t\equiv\widehat{t}\pmod{2}$ and show

[TABLE]

Note that the proof is equivalent when $t-1\equiv\widehat{t}\pmod{2}$.

Using Fact 5.3, we have the explicit probability

[TABLE]

Substituting $t$ by $t-2$ into the equation above and comparing the right hand sides gives

[TABLE]

because we know by assumption that

[TABLE]

Inductively applying this argument inductively for $t-2$ proves the inequality.

To complete our proof, it now suffices to show

[TABLE]

which we prove similarly. First rewrite the right hand side using the fact that

[TABLE]

and initially assume that $t\equiv n\pmod{2}$, which implies $n\equiv n-i\pmod{2}$. Again, using the explicit formula from Fact 5.3 and substituting $n$ by $n-2$ gives

[TABLE]

when $t+2\leq n^{2}$, which true by assumption and can be inductively applied until $n=(n-i+2)$ because $(n-i+2)\geq\lceil n/2\rceil+1$. Unfortunately, we cannot entirely apply the same proof when $t-1\equiv n\pmod{2}$ because this implies $n\not\equiv n-i\pmod{2}$. Applying the same proof as for $t\equiv n\pmod{2}$ we can obtain

[TABLE]

because $(t-1)+2\leq(n-i+3)^{2}$.

Therefore, we can conclude the proof by showing

[TABLE]

This is then true when

[TABLE]

which holds for $n-i\geq 0$. ∎

An immediate corollary of this Lemma 5.8 is that if we condition on the walk not going to the left of $1$, it only becomes more probable to reach $n$ without going above $n$. Now we prove the main result of this section. See 3.3

Proof.

Consider any starting position $1\leq i\leq\lceil n/2\rceil$ and any time $n^{2}/c\leq t\leq n^{2}/4$ with $t\equiv n-i\pmod{2}$. By the definition of conditional probability we have

[TABLE]

Lemma 5.6 shows that the second term is at least $\exp(-1-c))i/n$. Taking the probability under ${\min_{\leq t}({{\mathit{w}}})}\geq 1$ (i.e., the complementary event of ${\min_{\leq t}({{\mathit{w}}})}<1$) in Lemma 5.8 allows us to upper bound the first term using Lemma 5.7 by

[TABLE]

Putting these together then gives

[TABLE]

which completes the proof. ∎

5.7 Upper Bounding the Final, Maximum, and Minimum Position

We begin by splitting every $t$ step walk in half, and instead consider the probability of each walk satisfying the given conditions. In order to give upper bounds of these probabilities, we will relax the requirements, allowing us to more easily relate the probabilities to previously known facts about one-dimensional walks that we proved in Section 5.3. Furthermore, by splitting the walk in half we now have to consider all possible midpoints in $[1,n]$.

Lemma 5.9**.**

For all integers $1\leq n\leq t$, we have

[TABLE]

Proof.

By subdividing the walk roughly in half, we consider all possible positions of a walk after half of its steps such that the walk satisfies the maximum and minimum conditions. The second half of the walk must end at $n$, which implies the maximum position of the walk must be at least $n$. Thus, the first half of the walk only needs to not go above $n$. Accordingly, we can write

[TABLE]

Removing conditions that the walks must satisfy cannot decrease the probability, so our upper bound follows. ∎

From Fact 5.3 we can obtain explicit expressions for each inner term of the summation, which we then simplify into a strong bound on the summation in the following lemma.

Lemma 5.10**.**

For all integers $1\leq n\leq t$, we have

[TABLE]

Proof.

Apply the upper bound from Lemma 5.9 and examine each inner term in the summation. By the symmetry of walks, there must be an equivalent number of $\lfloor t/2\rfloor$ step walks with endpoints $1$ and $i$ that never walk below $1$ versus those that never walk above $i$. Thus

[TABLE]

Shifting the start of the walk to [math] allows us to apply Fact 5.3, because ${\max_{\leq\lfloor\frac{t}{2}\rfloor}({{\mathit{w}}})}\leq i$ is equivalent to ${\max_{\leq\lfloor\frac{t}{2}\rfloor}({{\mathit{w}}})}=i$ if the walk must end at $i$. Therefore,

[TABLE]

when the parity is correct and [math] otherwise. This works as an upper bound. Similarly, by shifting the start to [math] and applying Fact 5.3 we have

[TABLE]

Applying Lemma 5.9, we now have expressions for the term inside the summation, so

[TABLE]

which we upper bound by the fact that $(\lfloor\frac{t}{2}\rfloor+i+1)(\lceil\frac{t}{2}\rceil+(n-i+1)+1)\geq t^{2}/4$. This completes the proof. ∎

The following lemma gives a upper bound for the inner expression from Lemma 5.10 by bounding the binomial coefficients with the central binomial coefficients and using Stirling’s approximation.

Lemma 5.11**.**

For any integer $1\leq in$, we have

[TABLE]

Proof.

Given that $1\leq i\leq n$, we can crudely upper bound $i(n-i+1)$ by $n^{2}$. Additionally, we can will use Stirling’s approximation on the central binomial coefficient to upper bound our binomial coefficients.

[TABLE]

and

[TABLE]

The exponential terms will cancel and

[TABLE]

giving our desired bound. ∎

The upper bound in Lemma 5.11 is not sufficient for $t$ that are asymptotically less than $n^{2}$, so for these $t$ we need to give a more detailed analysis. Therefore, we more carefully examine the binomial coefficients that are significantly smaller than the central coefficient for small $t$. Consequently, the exponential term will not be sufficiently canceled by the binomial coefficient for values of $t$ that are asymptotically smaller than $n^{2}$. More specifically, we show that the function of $t$ on the right hand side of Lemma 5.10 is increasing in $t$ up until approximately $n^{2}$.

In the following lemma we consider even length walks for simplicity. The proof for odd length walks follows analogously.

Lemma 5.12**.**

For $n\geq 20$ and any integer $1\leq i\leq n$, for all $t\leq n^{2}/40$ we have

[TABLE]

where we consider walks of length $2t$ and $2t+4$ to ensure that $(2t)/2$ and $(2t+4)/2$ have the same parity.

Proof.

Canceling like terms implies that the desired inequality is equivalent to

[TABLE]

Examining the binomial coefficients shows that

[TABLE]

and

[TABLE]

Using these identities, our desired inequality equals

[TABLE]

Further cancellation of like terms and moving the denominator on each side into the numerator on the other side implies that our desired inequality is equivalent to

[TABLE]

It is straightforward to see that

[TABLE]

is maximized by $i=1$ and

[TABLE]

is maximized by $n-i=0$. Furthermore, it must be true that either $i\geq n/2$ or $n-i\geq n/2$, so we can upper bound the left hand side of our inequality by substituting $n/2$ for $i$ or $n-i$, and setting the other terms to the value that maximizes the product. Hence,

[TABLE]

To prove our desired inequality it now suffices to show $(t+2)^{2}\left(t+3+n/2\right)\left(t+3-n/2\right)\leq t^{2}(t+1)^{2}$, which is equivalent to

[TABLE]

Expanding both sides of the inequality and rearranging terms yields

[TABLE]

Given that $2t^{2}/(t+2)\leq 2t$, it suffices to show that $8t+9\leq n^{2}/4,$ which is true when $t\leq n^{2}/40$ and $n\geq 20$. ∎

We can now prove the main upper bound result of this section using the recently developed bounds for the right hand side of the expression in Lemma 5.10.

See 4.1

Proof.

Applying Lemmas 5.10 and 5.11 gives

[TABLE]

which immediately gives the upper bound $64(n/t)^{3}$. Similarly, Lemmas 5.10 and 5.12 imply that for $t\leq n^{2}/40$,

[TABLE]

where $T=n^{2}/40$. We then use Lemma 5.11 and sum from $1$ to $n$ to obtain

[TABLE]

for all $t\leq n^{2}/40$. Using the fact that $64(n/t)^{3}$ is a decreasing function in $t$, we have

[TABLE]

for all $t\geq n^{2}/40$, which is the desired bound. ∎

6 Extension to Higher Dimensions

Now we show how to extend our analysis to upper and lower bound the transience class of $d$-dimensional grids.

See 1.4

We denote by \text{d-}\textsc{Cube}_{n} the $d$-dimensional hypercube grid with $n^{d}$ vertices, and construct it analogously to $\textsc{Square}_{n}$. Its vertex set is $\{1,2,\dots,n\}^{d}\cup\{v_{\textnormal{sink}}\}$ and its edges connect any pair of vertices that differ in one coordinate. Vertices on the boundary have additional edges connecting to $v_{\textnormal{sink}}$ so that every non-sink vertex has degree $2d$. We use the vector notation $\boldsymbol{\mathit{u}}=(\boldsymbol{\mathit{u}}_{1},\boldsymbol{\mathit{u}}_{2},\dots,\boldsymbol{\mathit{u}}_{d})$ to identify non-sink vertices. We can decouple a walk $w$ on \text{d-}\textsc{Cube}_{n} into one-dimensional walks $w_{1},w_{2},\dots,w_{d}$, so that each step of a random walk on \text{d-}\textsc{Cube}_{n} can be understood as choosing a random direction with probability $1/d$ and then a step in the corresponding one-dimensional walk with probability $1/2$.

Our bounds for the two-dimensional grid heavily relied on decoupling walks into interleaved one-dimensional walks, and applying bounds from Section 5 for simple symmetric walks. Generalizing these bounds to $d$-dimensional hypercubes follows comparably and only requires simple extensions of our lemmas for the two-dimensional grids. Therefore, we will reference the necessary lemmas from previous sections and show the minor modifications needed to give analogous lemmas for the $d$-dimensional grid. The upper bound proof requires several key lemmas and is more involved, whereas extending the lower bound only requires one simple addition to our proof in Section 4.

6.1 Upper Bounding the Transience Class

Since Theorem 2.4 from [CV12] relies on non-sink vertices having constant degree, our assumptions that $d$ is constant and that all non-sink vertices have degree $2d$. In addition to utilizing properties of one-dimensional walks, specifically Lemma 3.3 proven in Section 5, the proof of our upper bound relies on four key lemmas:

• •

Lemma 2.9 — The source vertex can be swapped with a any non-sink vertex while only losing a $O(\log{n})$ approximation factor in the potential.

• •

Lemma 3.1 — An upper bound on the sum of all vertex potentials by factoring the expression into one-dimensional vertex potentials.

• •

Lemma 3.2 — For any vertex, the opposite corner vertex minimizes the potential up to a constant.

• •

Lemma 3.5 — A lower bound on the vertex potential $\boldsymbol{\pi}_{(n,n)}\left(\boldsymbol{\mathit{u}}\right)$, for any $\boldsymbol{\mathit{u}}$ in the top-right quadrant of $\textsc{Square}_{n}$.

Now we describe how to extend each of these lemmas to constant dimensions. These results almost immediately follow from decoupling walks into one-dimensional walks.

Lemma 6.1**.**

For any pair of non-sink vertices $\boldsymbol{\mathit{u}}$ and $\boldsymbol{\mathit{v}}$ in \text{d-}\textsc{Cube}_{n}, we have

[TABLE]

Proof.

This is a consequence of Rayleigh’s monotonicity theorem. Fix an underlying $n\times n$ subgraph of the hypercube with corners at the source and sink, and set the rest of the resistors to infinity. The upper bound for the $n\times n$ grid is an upper bound for the hypercube. ∎

Our lemma analogous to Lemma 3.1 follows from Lemma 6.1 and decoupling walks into one-dimension.

Lemma 6.2**.**

For any non-sink vertex $\boldsymbol{\mathit{u}}$ in \text{d-}\textsc{Cube}_{n},

[TABLE]

Proof.

Follow the proof structure of Lemma 3.1. ∎

We can also generalize our proof of Lemma 3.2 to higher dimensions, because we work with each dimension independently.

Lemma 6.3**.**

If $\boldsymbol{\mathit{u}}$ is a non-sink vertex of \text{d-}\textsc{Cube}_{n} such that $1\leq\boldsymbol{\mathit{u}}_{i}\leq\lceil n/2\rceil$ for all $1\leq i\leq d$, then

[TABLE]

Proof.

Extend the proof of Lemma 3.2 by reflecting walks across the $(d-1)$-dimensional hyperplane perpendicular to the chosen axis instead of a line. ∎

Lastly, we generalize Lemma 3.5, where the key idea was to considers walks of length $\Theta(n^{2})$ and show that there is a constant fraction such that both dimensions have taken $\Theta(n^{2})$ steps, which allows us to apply Lemma 3.3 for each possible walk. To do this, we essentially union bound Lemma 3.4 over $d$ dimensions, which shows that $\Theta(n^{2})$ walk lengths take $\Theta(n^{2})$ steps in each direction with probability at least $2^{-d}$.

Lemma 6.4**.**

For $n\geq 10$ and \boldsymbol{\mathit{u}}\in V(\text{d-}\textsc{Cube}_{n}) such that $1\leq\boldsymbol{\mathit{u}}_{i}\leq\lceil n/2\rceil$ for $1\leq i\leq d$, we have

[TABLE]

Proof.

Decouple walks ${\mathit{w}}\in\mathcal{W}(\boldsymbol{\mathit{u}}\rightarrow(n,n,\dots,n))$ into one-dimensional walks ${\mathit{w}}_{i}\in\mathcal{W}^{\textsc{Line}}(\boldsymbol{\mathit{u}}_{i})$, and view $\boldsymbol{\pi}_{(n,n,\dots,n)}\left(\boldsymbol{\mathit{u}}\right)$ as the probability that each walk ${\mathit{w}}_{i}$ is present on $n$ at the same time before any leaves the interval $[1,n]$. If each walk takes $t_{1},t_{2},\dots,t_{d}$ steps, respectively, then the total number of possible interleavings of these walks is the multinomial

[TABLE]

Just as before, we can obtain the lower bound

[TABLE]

To apply Lemma 3.3 to each walk, we need each $t_{i}$ to be in the interval $[n^{2}/c,n^{2}/4]$, for $c=16d$. Then we consider all walks of length

[TABLE]

where $t=t_{1}+t_{2}+\cdots+t_{d}$, and show that a constant fraction of these walks satisfy $t_{i}\geq n^{2}/c$ with $t_{i}$ having the correct parity. Note that we can ignore the parity conditions by simply lower bounding the probability of all having correct parity by $4^{-d}$. It then remains to show that all walks satisfy the inequality $t_{i}\geq n^{2}/c$ with constant probability.

Consider the probability that $t_{1}\geq n^{2}/c$. The other dimensions follow identically. Letting each dimension take at least $n^{2}/c$ steps introducing dependence, so we instead consider the probability that $t_{1}\geq n^{2}/c$ and condtion on $t_{2},t_{3},\dots,t_{d}\geq n^{2}/c$ (which can only decrease the probability of the event $t_{1}\geq n^{2}/c$). This is equivalent to fixing $n^{2}/c$ steps in each of those directions and randomly choosing all remaining steps with probability $1/d$ for each direction. The remaining number of steps is then at least $dn^{2}/c$ by our assumption that $t\geq n^{2}/8$. Therefore, the expected number of steps in the first dimension is at least $n^{2}/c$, which implies $t_{1}\geq n^{2}/c$ with probability at least $1/2$. Multiplying this probability over all dimensions gives $t_{i}\geq n^{2}/c$ with probability at least $2^{-d}$.

Thus, there are $O(n^{2})$ values of $t$ that we can decompose into one-dimensional walks, each occurring with constant probability. Applying Lemma 3.3 to each decomposition and summing

[TABLE]

over $O(n^{2})$ possible walks proves the claim. ∎

Now we prove \textnormal{tcl}(\text{d-}\textsc{Cube}_{n})=O(n^{3d-2}\log^{d+2}{n}) using Theorem 2.4. For any $\boldsymbol{\mathit{u}}=(\boldsymbol{\mathit{u}}_{1},\boldsymbol{\mathit{u}}_{2},...,\boldsymbol{\mathit{u}}_{d})$ in the top-left orthant of \text{d-}\textsc{Cube}_{n}, it follows that

[TABLE]

6.2 Lower Bounding the Transience Class

Extending our lower bound to $d$-dimensional hypergrids is a simple consequence of decoupling $d$-dimensional walks into one-dimensional walks, because we only need to generalize the upper bound in Lemma 4.3 to

[TABLE]

by replacing the negative binomial distribution with the negative multinomial distribution.

Fact 6.5**.**

For any nonnegative integer $t_{1}$, we have

[TABLE]

Proof.

Consider the proof of Fact 4.2 using the negative multinomial distribution. ∎

Thus, we can apply Lemma 4.1 and Lemma 4.4 to show

[TABLE]

By Theorem 2.4, we have \textnormal{tcl}(\text{d-}\textsc{Cube}_{n})=\Omega(n^{3d-2}).

Acknowledgments

We thank Marcel Celaya, Wenhan Huang, Wenhao Li, Pinyan Lu, Richard Peng, Dana Randall, and Chi Ho Yuen for various helpful discussions. We also thank the anonymous reviewers for their insightful comments, especially those that pointed out a major issue with Lemma 3.5 in a previous version of this manuscript.

Appendix A Appendix for Section 2

In this section, we prove a relationship between voltage potentials and the probability of a random walk escaping at the source instead of the sink.

See 2.6

Proof.

By definition, we have

[TABLE]

For any $\boldsymbol{\mathit{v}}\in V(\textsc{Square}_{n})$, let

[TABLE]

be the normalizing constant for $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}\left(\boldsymbol{\mathit{v}}\right)$. It follows that $f(\boldsymbol{\mathit{u}})=1$ and $f(v_{\textnormal{sink}})=1$, because the only such walk for each has length 0. For all other $\boldsymbol{\mathit{v}}\in V(\textsc{Square}_{n})\setminus\{\boldsymbol{\mathit{u}},v_{\textnormal{sink}}\}$, we have

[TABLE]

Therefore, $f(\boldsymbol{\mathit{v}})$ is a harmonic function with constant boundary values, so $f(\boldsymbol{\mathit{v}})=1$ for all vertices $\boldsymbol{\mathit{v}}\in V(\textsc{Square}_{n})$. ∎

We also verify that the effective resistance between $v_{\textnormal{sink}}$ and any internal vertex is bounded between $\Omega(1)$ and $O(\log n)$ using a triangle inequality for effective resistances and the fact that the effective resistance between opposite corners in an $n\times n$ resistor network is $\Theta(\log n)$. This proof easily generalizes to any pair of vertices in $\textsc{Square}_{n}$.

Proposition A.1** ([LPW09]).**

Let $G$ be an $n\times n$ network of unit resistors. If $u$ and $v$ are vertices at opposite corner vertices, then $\log(n-1)/2\leq\mathcal{R}_{\textnormal{eff}}\left(u,v\right)\leq 2\log n.$

See 2.8

Proof.

We first prove the lower bound

[TABLE]

The effective resistance between $v_{\textnormal{sink}}$ and $\boldsymbol{\mathit{u}}$ is the reciprocal of the total current flowing into the circuit when $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{u}})=1$ and $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(v_{\textnormal{sink}})=0$. Since $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}$ is a harmonic function, we have $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{v}})\geq 0$ for all $\boldsymbol{\mathit{v}}\in V(\textsc{Square}_{n})$. Moreover, $\text{deg}(\boldsymbol{\mathit{u}})=4$, so

[TABLE]

For the upper bound, we use Rayleigh’s monotonicity law, Proposition A.1, and the triangle inequality for effective resistances to show that

[TABLE]

for $n$ sufficiently large. Rayleigh’s monotonicity law [DS84] states that if the resistances of a circuit are increased, the effective resistance between any two points can only increase. The following triangle inequality for effective resistances is given in [Tet91]:

[TABLE]

Define $H$ to be the subgraph of $\textsc{Square}_{n}$ obtained by deleting $v_{\textnormal{sink}}$ and all edges incident to $v_{\textnormal{sink}}$. Let $m$ be the largest positive integer such that $\boldsymbol{\mathit{u}}_{1}+i\leq n$ and $\boldsymbol{\mathit{u}}_{2}+j\leq n$ for all $0\leq i,j<m$, and let $H(\boldsymbol{\mathit{u}})$ be the subgraph of $H$ induced by the vertex set

[TABLE]

We can view $H(\boldsymbol{\mathit{u}})$ as the largest square resistor network in $H$ such that $\boldsymbol{\mathit{u}}$ is the top-left vertex. Let $\boldsymbol{\mathit{v}}=[\boldsymbol{\mathit{u}}_{1}+m-1,\boldsymbol{\mathit{u}}_{2}+m-1]$ be the bottom-right vertex in $H(\boldsymbol{\mathit{u}})$. Using infinite resistors to remove every edge in $E(\textsc{Square}_{n})\setminus E(H(\boldsymbol{\mathit{u}}))$, we have

[TABLE]

by Rayleigh’s monotonicity law. Proposition A.1 implies that

[TABLE]

since $m\leq n$. The vertex $\boldsymbol{\mathit{v}}$ is incident to $v_{\textnormal{sink}}$ in $\textsc{Square}_{n}$, so Rayleigh’s monotonicity law gives

[TABLE]

By the triangle inequality for effective resistances, we have

[TABLE]

which completes the proof. ∎

Appendix B Appendix for Section 3

We use the random walk interpretation of voltage to prove Lemma 3.2. The key idea is that the voltage on the boundary opposite of $\boldsymbol{\mathit{u}}$ along any axis is less by a constant factor. This projection can be iterated along an axis in each dimension.

See 3.2

Proof.

We use Lemma 2.6 to decompose $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{v}})$ as a sum of probabilities of walks, and then construct maps for all $1\leq\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2}\leq n$ to show

[TABLE]

We begin by considering the first dimension:

[TABLE]

Let $\ell_{\text{hor}}$ be the horizontal line of reflection passing through $(\lceil(\boldsymbol{\mathit{v}}_{1}+n)/2\rceil,1)$ and $(\lceil(\boldsymbol{\mathit{v}}_{1}+n)/2\rceil,n)$ in $\mathbb{Z}^{2}$, and let $\boldsymbol{\mathit{u}}^{*}$ be the reflection of $\boldsymbol{\mathit{u}}$ over $\ell_{\text{hor}}$. Note that $\boldsymbol{\mathit{u}}^{*}$ may be outside of the $n\times n$ grid. Next, define the map

[TABLE]

as follows. For any walk $w\in\mathcal{W}((n,\boldsymbol{\mathit{v}}_{2})\rightarrow\boldsymbol{\mathit{u}})$:

Start the walk $f(w)$ at $(\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})$, and if $n-\boldsymbol{\mathit{v}}_{1}$ is odd move to $(\boldsymbol{\mathit{v}}_{1}+1,\boldsymbol{\mathit{v}}_{2})$. 2. 2.

Perform $w$ but make opposite vertical moves before the walk hits $\ell_{\text{hor}}$, so that the partial walk is a reflection over $\ell_{\text{hor}}$. 3. 3.

After hitting $\ell_{\text{hor}}$ for the first time, continue performing $w$, but now use the original vertical moves. 4. 4.

Terminate this walk when it first reaches $\boldsymbol{\mathit{u}}$.

Denote the preimage of a walk $w^{\prime}\in\mathcal{W}((\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})\rightarrow\boldsymbol{\mathit{u}})$ under $f$ to be

[TABLE]

We claim that for any $w^{\prime}\in\mathcal{W}^{\textsc{Square}_{n}}((\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})\rightarrow\boldsymbol{\mathit{u}})$,

[TABLE]

If $f^{-1}(w^{\prime})=\emptyset$ the claim is true, so assume $f^{-1}(w^{\prime})\neq\emptyset$. We analyze two cases. If $w^{\prime}$ hits $\ell_{\text{hor}}$, then $f^{-1}(w^{\prime})$ contains exactly one walk $w$ of length $|w^{\prime}|$ or $|w^{\prime}|-1$. If $w^{\prime}$ does not hit $\ell_{\text{hor}}$, then

[TABLE]

It follows that any walk $w\in f^{-1}(w^{\prime})$ can be split into $w=w_{1}w_{2}$, where $w_{1}$ is the unique walk from $(n,\boldsymbol{\mathit{v}}_{2})$ to $\boldsymbol{\mathit{u}}^{*}$ that is a reflection of $w^{\prime}$, and $w_{2}$ is a walk from $\boldsymbol{\mathit{u}}^{*}$ to $\boldsymbol{\mathit{u}}$ that avoids $v_{\textnormal{sink}}$ and hits $\boldsymbol{\mathit{u}}$ exactly once upon termination. Clearly $w_{1}$ has length $|w^{\prime}|$ or $|w^{\prime}|-1$, and the set of admissible $w_{2}$ is $\mathcal{W}(\boldsymbol{\mathit{u}}^{*}\rightarrow\boldsymbol{\mathit{u}})$. Therefore,

[TABLE]

since $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}(\boldsymbol{\mathit{u}}^{*})$ is an escape probability. Summing over all $w^{\prime}\in\mathcal{W}((\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})\rightarrow\boldsymbol{\mathit{u}})$, it follows from Lemma 2.6 and the previous inequality that

[TABLE]

because every $w\in\mathcal{W}((n,\boldsymbol{\mathit{v}}_{2})\rightarrow\boldsymbol{\mathit{u}})$ is the preimage of a $w^{\prime}\in\mathcal{W}((\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})\rightarrow\boldsymbol{\mathit{u}})$.

Similarly, we can show that $\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}\left((\boldsymbol{\mathit{v}}_{1},\boldsymbol{\mathit{v}}_{2})\right)\geq\boldsymbol{\pi}_{\boldsymbol{\mathit{u}}}\left((\boldsymbol{\mathit{v}}_{1},n)\right)/4$ for all $1\leq\boldsymbol{\mathit{v}}_{1}\leq n$ by reflecting walks over the vertical line from $(1,\lceil(n+\boldsymbol{\mathit{v}}_{2})/2\rceil)$ to $(n,\lceil(n+\boldsymbol{\mathit{v}}_{2})/2\rceil)$. Combining inequalities proves the claim. ∎

Lastly, we give a constant lower bound for the probability of an $n$-step simple symmetric walk being sufficiently close to its starting position by using the recursive definition of binomial coefficients and a Chernoff bound for symmetric random variables.

See 3.4

Proof.

First observe that for $n\geq 10$, we have

[TABLE]

and

[TABLE]

To see this, use the parity restriction and expand the summands as

[TABLE]

Let $X_{1},X_{2},\dots,X_{n-1}$ be independent Bernoulli random variables such that $\mbox{{Pr}}\left[{X_{i}=0}\right]=1/2$ and $\mbox{{Pr}}\left[{X_{i}=1}\right]=1/2$. Let $S_{n-1}=X_{1}+X_{2}+\dots+X_{n-1}$ and $\mu=E[S_{n-1}]=(n-1)/2$. Using a Chernoff bound, we have

[TABLE]

for $n\geq 60$. Checking the remaining cases numerically when $10\leq n<60$ proves the claim. ∎