The range of once-reinforced random walk in one dimension

TL;DR

This paper investigates the properties of once-reinforced random walk on the integer lattice, deriving comprehensive limit results for all moments of its range through advanced Tauberian techniques.

Contribution

It provides the first detailed moment analysis of the range of ORRW in one dimension using Tauberian theory, expanding understanding of reinforced random walks.

Findings

Derived limit results for all moments of the range

Applied Tauberian theory to reinforced random walk

Enhanced theoretical understanding of ORRW behavior

Abstract

We study once-reinforced random walk (ORRW) on $\mathbb Z$. For this model, we derive limit results on all moments of its range using Tauberian theory.

Equations

$$Y_t=B_t+(1-c)(\sup \{Y_s:s\le t\}+|\inf \{Y_s:s\le t\}|).$$

$$X_{n+1}=X_n+1\text{ with probability }=\{\begin{array}{ll}\frac{1}{1+c}& \text{ if }{X}_n=M_n\\ \frac{1}{2}& \text{ if }{X}_n\ne m_n,M_n,\\ \frac{c}{1+c}& \text{ if }{X}_n=m_n\end{array}$$

$$\text{and }\mathbb{P}(X_{n+1}=X_n-1)=1-\mathbb{P}(X_{n+1}=X_n+1).$$

$$R_n:=M_n-m_n=M_n+|m_n|.$$

$$S_k=1+\sum _{i=1}^{k-1}{T}_i,k=1,2,...\text{ with }{T}_i=1+\sum _{j=1}^{Y_i}\left(1+{\tau }_i^j\right),i=1,2,...$$

$$\displaystyle\mathbb{E}\Big{[}\Big{(}\frac{R_{n}}{\sqrt{n}}\Big{)}^{\ell}\Big{]}$$

$$J_{\ell }(c)$$

$$\displaystyle\mathbb{E}\Big{[}\frac{R_{n}}{\sqrt{n}}\Big{]}$$

$$\displaystyle\mathbb{E}\Big{[}\frac{R_{n}}{\sqrt{n}}\Big{]}\sim\sqrt{\frac{8}{\pi}},\qquad\qquad\mathbb{E}\Big{[}\Big{(}\frac{R_{n}}{\sqrt{n}}\Big{)}^{2}\Big{]}\sim 4\log 2.$$

$$J_1(1)$$

$$J_2(1)$$

$$\stackrel{y=e^x}{=}4{\int }_1^{\infty }\frac{1}{y}-\frac{1}{y+1}dy=4{\int }_1^2\frac{1}{y}dy=4\log 2,$$

$$\mathbb{E}[S_k]=\left(\genfrac{}{}{0ex}{}{k+1}{2}\right),\mathbb{V}[S_k]=\frac{(k-1)k(k+1)(k+2)}{12}.$$

$$S_k=\sum _{i=0}^{k-1}{\tau }_{i+2},$$

$$\frac{d}{dx}\sum_{j=0}^{c-1}\binom{c-1}{j}(-1)^{c-1-j}\frac{1}{j-2c+1}(e^{x}+1)^{j-2c+1}=\Big{(}\frac{e^{x}}{(e^{x}+1)^{2}}\Big{)}^{c}.$$

$$J_1(c)$$

$$\displaystyle\frac{d}{dx}\Big{(}x$$

$$J_2(c)$$

$$=2^{2c}\sum _{j=0}^{c-1}\left(\genfrac{}{}{0ex}{}{c-1}{j}\right)(-1{)}^{c-j}\frac{1}{j-2c+1}{\int }_0^{\infty }\frac{1}{(e^x+1{)}^{2c-j-1}}dx$$

$$\displaystyle=2^{2c}\sum_{j=0}^{c-1}\binom{c-1}{j}(-1)^{c-j}\frac{1}{j-2c+1}\Big{(}x-\log(e^{x}+1)+\sum_{i=1}^{2c-j-2}\frac{1}{i(e^{x}+1)^{i}}\Big{)}\Big{|}_{x=0}^{\infty}$$

$$\displaystyle=2^{2c}\sum_{j=0}^{c-1}\binom{c-1}{j}(-1)^{c-j-1}\frac{1}{2c-j-1}\Big{(}\log(2)-\sum_{i=1}^{2c-j-2}\frac{1}{i2^{i}}\Big{)}.$$

$$X_{n.}/\sqrt{n}\stackrel{n\to \infty }{\Rightarrow }Y=(Y_t{)}_{t\ge 0},$$

$$\mathbb{E}[{\hat{R}}_t^{\ell }]$$

$$Y_t-(1-c)(\sup \{Y_s:s\le t\}+|\inf \{Y_s:s\le t\}|)\text{ is a Brownian motion.}$$

$$N_n$$

$$N_{n+1}^1-N_n^1$$

$$\langle N\rangle_{nt}=nt-\Big{(}\frac{1-c}{1+c}\Big{)}^{2}\Big{(}\sum_{k<nt}1_{X_{k}=M_{k}}+\sum_{k<nt}1_{X_{k}=m_{k}}\Big{)}\sim nt-\frac{(1-c)^{2}}{1+c}R_{nt},$$

$$\Big{(}\frac{X_{nt}}{\sqrt{n}}-(1-c)\Big{(}\frac{M_{nt}}{\sqrt{n}}+\frac{|m_{nt}|}{\sqrt{n}}\Big{)}\Big{)}_{t\geq 0},$$

$$N_n$$

$$N_{n+1}^1-N_n^1$$

$$2\sum _{k<n}\mathbb{E}[M_k{1}_{X_k=M_k\}}]$$

$$\displaystyle=(1+c)\cdot\mathbb{E}\Big{[}\Big{(}\sum_{k<n}1_{X_{k}=M_{k}}\Big{)}^{2}\Big{]}+O(\sqrt{n})=\frac{1}{1+c}M_{n}^{2}+O(\sqrt{n}),$$

$$\mathbb{E}[X_n^2]$$

$$\frac{R_n^2}{2}=\frac{M_n^2+2M_n|m_n|+|m_n{|}^2}{2}\le M_n^2+|m_n{|}^2\le R_n^2,$$

Full text

††AMS 2000 subject classification. 60J15 (Primary) , 60K99 (Secondary).††Keywords and phrases. Reinforced random walk; range process; scaling limit

1 Introduction

Interacting random walks are a class of (mostly non-Markovian) models where the next step of (mostly simple) random walk depends on its path. Some models tend to visit new sites with high probability. Such models arose in chemical physics as a model for long polymer chains, and are discussed in some detail in Chapter 6 of Lawler (2013). A particularly prominent example is the self-avoiding walk, which visits every site not more than once. Other models are the myopic (or true) self-avoiding walk, which has higher chances to move to sites it visited less than others. Another class of random walk models – usually referred to as reinforced random walks – visit sites (or walk along edges) more likely they have already seen; see e.g. Davis (1990) and Pemantle (2007).

Here, we study a model which appeared as the hungry random walk in the Physics literature (Schilling and Voigtmann, 2017) for mimicking chemotaxis. For the model in $d=1$, every site in $\mathbb{Z}$ contains food, which is eaten up once the walker visits the site. In addition, the walker rather visits sites with food, if it sees a neighboring site containing food (with probability proportional to $e^{\gamma}$ for some $\gamma\in\mathbb{R}$) than going the other direction (with probability proportional to 1). Two models from the probability literature are equivalent to the hungry random walk on $\mathbb{Z}$: First, the true self-avoiding walk with generalized bond-repulsion, as studied in Toth (1993), jumps along an edge $b$ with probabilities proportional to \exp(-\gamma\cdot(\text{number of previous jumps along b})^{\kappa}), which equals the hungry random walk in the limit $\kappa\to 0$. Second, the case $\gamma<0$ is equivalent to once-reinforced random walk or ORRW: here, every edge has initial weight 1, and once the walker goes along an edge, the weight changes to $c>1$. The walker then chooses its next step according to the edge weights. Apparently, this is the same as the hungry random walker for $\gamma=-\log c<0$ on $\mathbb{Z}$. Recent literature on the ORRW has focused on recurrence and transience on various graphs (see e.g. Durrett et al., 2002; Dai, 2005; Sellke, 2006; Kious and Sidoravicius, 2018). Here, we rather stick to $\mathbb{Z}$ but aim at concrete formulas for the asymptotics of the range of ORRW in Theorem 1. Our analysis is based on a simple decomposition of the inverse of the range process as given in (3). Notably, we cannot compute moments of ORRW itself. At least, we give some heuristics of the variance in Remark 2.5.

In studying the ORRW, we will not restrict ourselves to $c>1$, but to $c>0$. A scaling limit of the ORRW in this case was studied in Davis (1996), Perman and Werner (1997) and Carmona et al. (1998). More precisely, it was shown (see Theorem 1.2 in Davis, 1996) that (choose $\alpha=-\beta=1-c$) for $1/2<c<3/2$, the sequence $X^{n}=(X_{nt}/\sqrt{n})_{t\geq 0}$ has a limit $Y$ as $n\to\infty$ which solves

[TABLE]

More connections of our results to this equation are discussed in Remark 2.4.

The paper is organized as follows: In the next Section, we give our main result, Theorem 1, which gives asymptotics of all moments of the range of the ORRW. Section 3.1 contains some preliminary steps for our proofs. The proof of Theorem 1 is given in Section 3.2.

2 Results

Definition 2.1**.**

Let $c>0$ and $X=(X_{n})_{n=0,1,2,...}$ be the stochastic process with $X_{0}=0$, and, for $n=0,1,2,...$, given $X_{0},...,X_{n}$, and setting $M_{n}:=\max_{k\leq n}X_{k}$ as well as $m_{n}:=\min_{k\leq n}X_{k}$,

[TABLE]

In other words, $X_{n+1}=X_{n}\pm 1$ with probability proportional to $c$ or $1$, if $X$ has or has not visited $X_{n}\pm 1$ before time $n$. We call $X=(X_{n})_{n=0,1,2,...}$ the once-reinforced random walk (or ORRW) on $\mathbb{Z}$ with parameter $c$. Its range by time $n$ is given by

[TABLE]

∎

Note that only the case $c>1$ gives a reinforced walk (in the sense that it visits previously seen sites more likely), while the walk has self-avoiding properties for $0<c<1$. For $c=1$, it is just the symmetric Bernoulli walk. Since our proofs work in all cases, we do not distinguish them in the sequel.

The range process $R=(R_{n})_{n=0,1,...}$ is a nondecreasing process with jumps of size 1, and is our main object of study. The following ideas are essential to understand our approach. The random time $S_{k}:=\inf\{n:R_{n}=k\}$ is the first time the ORRW has range $k$ (such that $k\mapsto S_{k}$ is the generalized inverse of $n\mapsto R_{n}$), and $T_{i}:=S_{i+1}-S_{i}$ is the time between $R_{n}=i$ for the first time and $R_{n}=i+1$. In order to study $T_{i}$ (for $i=1,2,...$), we note that $T_{i}=1$ with probability $1/(1+c)$. Otherwise, the random walk moves within its range (which is $i$ at that time) until it first hits its maximum or minumum, which takes time $\tau_{i}$, the hitting time of $\{-1,i-1\}$ of a simple random walk starting in 0. Again, the chance to increase the range is $1/(1+c)$ etc. The number of times the random walk needs a chance of $1/(1+c)$ to increase its range is a geometrically distributed random variable $Y_{i}$ with parameter $1/(1+c)$. (Note that $Y_{i}=0$ is possible, i.e. we must use the shifted geometrical distribution.) In total, this gives

[TABLE]

(where we define the empty sum to be 0). Here, $\tau_{i}^{k},k=1,2,...$ are independent and identically distributed as $\tau_{i}$ above and also independend from $Y_{i},i=1,2,...$ Using (3), we can compute the generating function of $S_{k}$, (see Lemma 3.2) and then use $\mathbb{P}(R_{n}>k)=\mathbb{P}(S_{k}<n)$ in order to obtain results on $R_{n}$ (see Lemma 3.3 and Proposition 3.4).

We are now ready to formulate our main result, which will be proved in Section 3.2. Throughout, we will write $a_{n}\sim b_{n}$ if $a_{n}/b_{n}\xrightarrow{n\to\infty}1.$

Theorem 1** (Asymptotic moments of the range).**

*Let $R_{n}$ be as in (2) the range of the ORRW with parameter $c>0$. Then, *

[TABLE]

In particular,

[TABLE]

Remark 2.2** (The range for $c=1$).**

The ORRW with $c=1$ equals the the symmetric Bernoulli walk. In this case, several results have been obtained for the range. An early example is Feller (1951), who states in his (1.4) that

[TABLE]

In this case, we compute

[TABLE]

and Feller’s result for the expectation follows from (4). Moreover, using integration by parts,

[TABLE]

which gives Feller’s result for the second moment.

In addition to these limiting results, Vallois (1993) and Vallois and Tapiero (1995) have computed the generating function as well as expectation and variance for $S_{k}$, given through

[TABLE]

These results can as well be obtained as follows: Modyfing (3) for the case $c=1$, we can write

[TABLE]

where $\tau_{i+2}$ is the hitting time of $\{-1,i+1\}$ of a random walk starting in 0. This holds since the range increases if and only if such a hitting time is observed. We note that $\tau_{i+2}$ is the duration of play of a symmetric Gambler’s ruin starting with 1 and a total of $i+2$ units. It is a classical result that $\mathbb{E}[\tau_{i+2}]=i+1$, and $\mathbb{V}[\tau_{i+2}]=\frac{(i+2)(i+1)i}{3}$ was e.g. derived in Bach (1997). Summing then gives Vallois’ results.

Remark 2.3** ($J_{\ell}(c)$ for integer-valued $c$).**

If $c$ is an integer, i.e. $c=1,2,...$, the calculations from (5) and (6) can be generalized and lead to specific expressions for $J_{\ell}(c)$. We just give the necessary steps for $\ell=1,2$, which can then be generalized for larger $\ell$. First, note that a straight-forward calculation shows that, for $c=1,2,...$

[TABLE]

Hence,

[TABLE]

Next,

[TABLE]

and therefore, using integration by parts,

[TABLE]

For higher $\ell$, more steps using integration by parts are necessary. ∎

Remark 2.4** (Scaling limit of ORRW).**

Theorem 1 can be understood as results for the scaling limit

[TABLE]

where $Y$ is given in (1); see Davis (1996) for the corresponding limit result. By this, we mean that the range $\hat{R}$ of $Y$ satisfies

[TABLE]

While the convergence above was only shown for $1/2<c<3/2$ in Davis (1996), we briefly argue how this converges comes about: Note that $Y$ solves (1) iff

[TABLE]

For the ORRW, note that

[TABLE]

For large $n$, we have that $\frac{1}{1+c}\sum_{k<nt}1_{X_{k}=M_{k}}\sim M_{nt}$ by the law of large numbers, since every time with $X_{k}=M_{k}$ there is an independent chance of $1/(1+c)$ of increasing $M$. Moreover, a straight-forward calculation gives that $N$ has quadratic variation

[TABLE]

where the $\sim$ follows from $\tfrac{1}{1+c}(\sum_{k<nt}1_{X_{k}=M_{k}}+\sum_{k<nt}1_{X_{k}=m_{k}})\sim R_{n}$ by the same argument using the law of large numbers as above. Since $\mathbb{E}[R_{nt}]=O(\sqrt{nt})$, as we have shown in Theorem 1, we thus have that the limit of $\frac{N_{n.}}{\sqrt{n}}$ as $n\to\infty$ is the same as the limit of

[TABLE]

which must be a continuous martingale with quadratic variation $t$ by time $t$, i.e. a Brownian motion. This is enough to conclude that scaling limits of $X$ satisfy (7).

Remark 2.5** (Towards $\mathbb{V}[X_{n}/\sqrt{n}]$).**

Although we are able to asymptotically compute all moments of $R_{n}/\sqrt{n}$ as in Theorem 1, we are unable to compute asymptotics of (even) moments of $X_{n}/\sqrt{n}$. At least, we now give some thoughts and bounds of asymptotics of $\mathbb{V}[X_{n}/\sqrt{n}]=\mathbb{E}[(X_{n}/\sqrt{n})^{2}]$. We observe that

[TABLE]

is a mean-zero-martingale, since (note that $m_{k}\leq 0\leq M_{k}$)

[TABLE]

Now, for large $n$, we have $M_{n}=(1+c)\sum_{k<n}1_{X_{k}=M_{k}}+o(\sqrt{n})$ by the law of large numbers, hence

[TABLE]

and by symmetry

[TABLE]

which gives the intuitive result that $\mathbb{E}[X_{n}^{2}]\leq n$ for $c>1$ and $\mathbb{E}[X_{n}^{2}]\geq n$ for $c<1$. Moreover, since $0\leq(M_{n}-|m_{n}|)^{2}$ implies $2M_{n}|m_{n}|\leq M_{n}^{2}+|m_{n}|^{2}$,

[TABLE]

it also gives the bounds

[TABLE]

From Figure 1, we see that the left hand side (LHS) performs better for $c>1$, while the right hand side (RHS) is better for $c<1$. The reason is that for $c>1$, the process is more likely to switch between its maximum and minimum, such that $M_{n}\approx|m_{n}|\approx R_{n}/2$, leading to $M_{n}^{2}+|m_{n}|^{2}\approx R_{n}^{2}/2$, while for $c<1$ switching becomes less likely and we rather have that $R_{n}\approx M_{n}$ or $R_{n}\approx|m_{n}|$, which gives $M_{n}^{2}+|m_{n}|^{2}\approx R_{n}^{2}$.

3 Proof of Theorem 1

3.1 Some preliminairies

Before we come to the proof of Theorem 1, we need some general results. First, in Theorem 2, we recall a classical Tauberian result by Hardy and Littlewood, which will help us to interprete the generating function of $S_{k}$ from (3). Then, in Lemma 3.1, we recall the generating function of hitting times for a simple symmetric random walk.

Theorem 2** (A Tauberian result).**

Let $a_{1},a_{2},...\geq 0$ such that $\sum_{n=1}^{\infty}a_{n}x^{n}$ converges for $|x|<1$. Suppose that for some $\alpha,A\geq 0$

[TABLE]

Then,

[TABLE]

Moreover, if $\alpha>1$, and $n\mapsto a_{n}$ is non-decreasing,

[TABLE]

Proof.

The assertions are classical Tauberian results by Hardy and Littlewood; see e.g. Chapter I.7.4 of Korevaar (2013). Another self-contained proof is given in Proposition 12.5.2 in Lawler and Limic (2010). ∎

The following lemma is rather standard (see e.g. Chapter XIV.4 in Feller, 1968), but we provide a proof for completeness.

Lemma 3.1** **(Generating function of hitting times in simple symmetric

random walk).

Let $Z$ by a simple symmetric random walk with $Z_{0}=0$ and $a,b>0$. Define $T_{k}:=\inf\{n:Z_{n}=k\}$, the first hitting time of $k\in\mathbb{Z}$, as well as $T:=T_{-a}\wedge T_{b}$ for $a,b\geq 0$. Then,

[TABLE]

where

[TABLE]

Proof.

Recall $\cosh(r):=\frac{e^{r}+e^{-r}}{2}$ and note that $\cosh(d_{s})=1/s$. For any $r\in\mathbb{R}$, using that

[TABLE]

the stochastic process $(e^{rZ_{n}}/\mathbb{E}[e^{rZ_{n}}])_{n=0,1,2,...}$ is a martingale. Therefore, using $r=\pm d_{s}$,

[TABLE]

is a martingale as well. We apply the optional sampling theorem to the bounded martingale $M_{T\wedge n}$ to obtain

[TABLE]

From this, we read off the result. ∎

3.2 Proof of Theorem 1

Our analysis of the moments of $R_{n}$ will be done via an analysis of the generating function of the random variable $S_{k}$, which we defined in (3). We start by computing the generating function of $S_{k}$.

Lemma 3.2** **(Generating function of $\tau_{i},T_{i}$ and

$S_{k}$).

Fix $s\in(0,1)$, recall $d_{s}$ from (9) and let

[TABLE]

Then, for $\tau_{i}$ and $T_{i}$ as in (3), the generating functions are given by

[TABLE]

Moreover, the generating function of $S_{k}$ is given by

[TABLE]

Proof.

The first claim follows directly from Lemma 3.1. For the generating function of $T_{i}$, note that the generating function of $Y_{i}\sim\text{geo}(1/(1+c))$ is

[TABLE]

hence

[TABLE]

The form of the generating function of $S_{k}$ follows from (3). ∎

Lemma 3.3** (A generating function for moments of $R_{n}$).**

Recall $G_{x}(s)$ from (10). Then, for $s\in(0,1)$ and $\ell=0,1,2,...$

[TABLE]

Proof.

We will use, for $k,n=1,2,...$

[TABLE]

Then,

[TABLE]

and the result follows from (11). ∎

Proposition 3.4** (Asymptotics of $H_{\ell}$).**

Let $\ell=0,1,2,...$ and $H_{\ell}$ as in Lemma 3.3. Then,

[TABLE]

where

[TABLE]

Proof.

We define

[TABLE]

and write for $g_{.}(s)$ as in (10), using the definition of $d_{s}$,

[TABLE]

as well as

[TABLE]

In the sequel, we will use $t:=t(s):=1-s$ throughout, set

[TABLE]

and note that

[TABLE]

as $t\to 0$. Therefore,

[TABLE]

The latter can now be used for (note that the empty product arising for $\ell=0$ is defined to be 1)

[TABLE]

where we have used two approximations of integrals with Riemann-sums (with $\sqrt{2t}\approx dx,dy$). The result follows from

[TABLE]

∎

Proof of Theorem 1.

We now combine the results of Proposition 3.4 with the Tauberian result from Theorem 2. We obtained in Proposition 3.4 for $a_{n}=\mathbb{E}[R_{n}\cdots(R_{n}+\ell)]$ that

[TABLE]

and we can apply Theorem 2 with $A=K_{\ell}$ and $\alpha=(3+\ell)/2$. In particular, (8) gives that

[TABLE]

Since $\Gamma(x+1)=x\Gamma(x)$, this implies

[TABLE]

and we are done. ∎

Acknowledgements

We thank Tanja Schilling for introducing us to the model of the hungry random walk.