Metastable Behavior of Bootstrap Percolation on Galton-Watson Trees

TL;DR

This paper investigates the metastable states in bootstrap percolation on Galton-Watson trees, revealing multiple metastable states with different durations depending on offspring distribution near criticality.

Contribution

It uncovers the existence of multiple metastable states in bootstrap percolation on Galton-Watson trees and characterizes their scaling behavior near criticality.

Findings

Multiple metastable states identified

Duration scaling varies with offspring distribution

Behavior depends on criticality proximity

Abstract

We analyze the metastable states near criticality of the bootstrap percolation on Galton-Watson trees. We find that, depending on the exact choice of the offspring distribution, it is possible to have several distinct metastable states, with varying scaling of their duration while approaching criticality.

Figures

Equations

$${\varphi }_{t+1}^d$$

$$h_d\left(x\right)$$

$${\varphi }_{t+1}^{\xi }$$

$$h_{\xi }\left(x\right)$$

$${\rho }_R\left(t\right)=\frac{\left|\left\{\text{mbox}infectedverticesinB(R)\text{mbox}attimet\right\}\right|}{\left|B(R)\right|}.$$

$$q_c=\underset{\left[0,1\right]}{\sup }\left\{q:{\varphi }_{\infty }^{\xi }=0\right\}.$$

$$g_k\left(x\right)$$

$$g_{\xi }\left(x\right)$$

$${\tau }_{x,\delta }^{-}(q)=\min \{t:{\varphi }_t^{\xi }<x+\delta \},$$

$${\tau }_{x,\delta }^{+}(q)=\min \{t:{\varphi }_t^{\xi }<x-\delta \}.$$

$$\frac{\log ({\varphi }_t^{\xi })}{\log t}\stackrel{t\to \infty }{\to }-\frac{1}{\nu },$$

$$g_{\xi }\left(x\right)=\frac{1}{q_c}-C_i{\left(x-x_i\right)}^{2{\nu }_i}+o\left({\left(x-x_i\right)}^{2{\nu }_i}\right),$$

$${\rho }_{R,w}\left(t\right)=\frac{\left|\left\{\text{mbox}infectedverticesinB\left(R\right)\setminus B\left(R-w\right)\text{mbox}attimet\right\}\right|}{\left|B\left(R\right)\setminus B\left(R-w\right)\right|}.$$

$${\rho }_R\left(t\right)={\rho }_{R,w}\left(t\right)+O\left(2^{-w}\right).$$

$$\mathbb{P}\left[\left|{\rho }_{r,1}(t)-\left(1-{\varphi }_t^{\xi }\right)\right|>\frac{\epsilon }{2}\right]\le e^{-c{2}^{R-w+1}}$$

$$\mathbb{P}\left[\left|{\rho }_{R,w}\left(t\right)-\left(1-{\varphi }_t^{\xi }\right)\right|>\frac{\epsilon }{2}\right]\le w{e}^{-c{2}^{R-w+1}}.$$

$$g_k\left(x\right)$$

$${\varphi }_{t+1}={\varphi }_t-C{q}_c{\varphi }_t^{\nu +1}+o({\varphi }_t^{\nu +1})$$

$$\frac{\text{d}\varphi }{\text{d}t}=-C{q}_c{\varphi }_t^{\nu +1}.$$

$$\frac{\text{d}\varphi }{\text{d}t}=-\frac{x_i}{q_c}(q_c-q)-C_i{q}_c{x}_i(\varphi -x_i{)}^{2{\nu }_i}.$$

$${\chi }_k=\{\begin{array}{ll}\frac{r-1}{k\left(k-1\right)}& r\le k\le \nu +r-1\\ 0& k\ge \nu +r\end{array}.$$

$$g_{\chi }\left(x\right)=1-\sum _{k=\nu +r}^{\infty }\frac{r-1}{k\left(k-1\right)}{g}_k\left(x\right).$$

$$P\left(x\right)=\sum _{k=\nu +r}^{\infty }\frac{r-1}{k\left(k-1\right)}\frac{g_k\left(x\right)}{x^{\nu }}.$$

$$\text{mbox}Pr{M}_{0}P=\text{mbox}Pr1.$$

$$\text{mbox}Pr{M}_{x_1,\dots ,x_n}\overline{P}=\text{mbox}Pr1$$

$$x_n=f(x_{n-1}).$$

$$y-\underline{c}{\left(y-y_0\right)}^{\alpha }\le f(y)\le y-\overline{c}{\left(y-y_0\right)}^{\alpha },$$

$$x_{n+1}-y_0$$

$$y_0+\underline{a}{\left(n+n_0\right)}^{-\frac{1}{\alpha -1}}\le x_n\le y_0+\overline{a}{n}^{-\frac{1}{\alpha -1}},$$

$$-(1-\alpha )\le \frac{{\left(1-z\right)}^{1-\alpha }-1}{z}\le -(1-\alpha )(1-\delta {)}^{-\alpha }.$$

$$x_n^{\epsilon }=f_{\epsilon }\left(x_{n-1}^{\epsilon }\right),$$

$$y-\underline{c}{\left(y-y_0\right)}^{2\alpha }-\epsilon \le f_{\epsilon }\left(y\right)\le y-\overline{c}{\left(y-y_0\right)}^{2\alpha }-\epsilon ,$$

Full text

1 Introduction

Bootstrap percolation is a deterministic dynamics in discrete time first introduced in [7] in order to model disordered magnetic systems, and broadly studied since in many different contexts. Fix a graph $G$ and a parameter $r\in\mathbb{N}$. Each vertex of the graph can be in one of two states – infected or healthy, which are initially distributed independently with probabilities $p$ and $q=1-p$. At each time step we update these states, such that the infected vertices remain infected, and a healthy vertex becomes infected if it has at least $r$ infected neighbors. One may also consider more general infection conditions, such as the oriented bootstrap percolation – when the graph $G$ is oriented, and we require at least $r$ edges to point at infected vertices.

Bootstrap percolation on various deterministic graphs has been the subject of extensive research. On the grid $\left[n\right]^{d}$, the probability that all vertices are eventually infected, as a function of $p$ (or equivalently $q$), has been profoundly studied in [1, 11, 2]. For $\left(d+1\right)$-regular infinite trees, with $2\leq r\leq d$, it is shown in [3] that a phase transition occurs. Defining $q_{c}$ to be the supremum over all $q$ such that starting with probability $q$ to be healthy all vertices end up being infected with probability $1$, an explicit expression for $q_{c}$ is found, and it is furthermore proven that $q_{c}$ lies in the open interval $\left(0,1\right)$. In addition, it is determined, depending on $d$ and on $r$, when the transition is continuous and when it is discontinuous. In [5] the details of the metastability properties are studied, describing the time evolution of the probability that the root stays healthy near criticality.

Random environments have also been of interest in this field, e.g., the bootstrap percolation on a polluted grid [10, 9], the random graph $G_{n,p}$ [13], the random regular graph [4, 12], and the Galton-Watson tree [6].

In this paper, we will analyze the metastability of the bootstrap percolation on a directed Galton-Watson tree, i.e., the time behavior near criticality of the probability that the root is infected. In 3.1 we present an interpretation of this probability as the almost sure prevalence – the limiting ratio of infected vertices. In 3.2 we will study the zoology of the metastabilities for different offspring distributions, showing that this model introduces a vast variety of possible behaviors. Finally, in 5 we comment on other phase transitions that may occur.

2 Model and Notations

Fix an infection threshold $r\geq 2$, and consider a Galton-Watson tree $G$ whose offspring distribution is supported on $r,r+1,\dots$ That is, defining $\xi_{k}$ to be the probability that a vertex has $k$ children, we require $\xi_{k}=0$ for $k<r$.

In the beginning, we decide for each vertex of $G$ whether it is infected or healthy, independently with probabilities $p$ and $q=1-p$ respectively. Then, at each time step $t$, a healthy vertex will get infected if it has at least $r$ infected children. Let us denote by $\phi_{t}^{G}$ the (random) probability that the root is healthy at time $t$, so in particular $\phi_{0}^{G}=q$. Note also that $\phi_{t}^{G}$ is decreasing in $t$. The expected value of $\phi_{t}^{G}$ over all graphs $G$, generated with offspring distribution $\xi$, will be denoted $\phi_{t}^{\xi}$.

One particular case, that has been studied in [3, 5, 8, 7], is the case of a rooted $\left(d+1\right)$-regular tree, i.e., $\xi_{k}=\mathds{1}_{k=d}$. Here, one can find $\phi_{t}^{d}$ recursively using the relation

[TABLE]

For the GW tree, such a recursion still holds for the expected value $\phi_{t}^{\xi}$:

[TABLE]

3 Results

3.1 Prevalence and $\phi_{t}$

The relation in (LABEL:recursion) allows us to find the expected value of $\phi_{t}^{G}$, but for a specific realization of $G$, $\phi_{t}^{G}$ may differ from that value. For example, fixing $t$, there is a nonzero probability that a finite neighborhood of the root will have many vertices of high degree, which will result in a smaller $\phi_{t}^{G}$. However, we will see that $\phi_{t}^{\xi}$ describes almost surely another observable – the prevalence, i.e., the limiting fraction of infected vertices.

First, denote by $B(R)$ the ball of radius $R$ around the root. We can then define the $R$-prevalence at time $t$ as

[TABLE]

It is natural to expect $\rho_{R}\left(t\right)$ to be close to $1-\phi_{t}^{\xi}$, and this is indeed the case, as shown in the following proposition:

Proposition 1**.**

Fix $t$. Then $\lim_{R\rightarrow\infty}\rho_{R}\left(t\right)=1-\phi_{t}^{\xi}$ almost surely (in both the graph and the initial state measures).

3.2 Critical Behavior

Following [3, 6], we define the critical probability

[TABLE]

In order to analyze this criticality, define

[TABLE]

In [6], the following fact is shown:

Fact 1**.**

Fix $\xi$. Then:

For a given $q$, $\phi_{\infty}^{\xi}$ is the maximal solution in $\left[0,1\right]$ of $g_{\xi}\left(x\right)=\frac{1}{q}$, and [math] if no such solution exists. 2. 2.

$q_{c}=\frac{1}{\max_{\left[0,1\right]}g_{\xi}\left(x\right)}$.

We will consider here the behavior near criticality, at $q$ slightly smaller than $q_{c}$.

Definition 1**.**

For $0<x<1$ and some positive $\delta$, the $\delta$-entrance time of $x$ is

[TABLE]

and the $\delta$-exit time is defined as

[TABLE]

Definition 2**.**

Fix $\delta>0$. We say that the critical point is $\delta$-$\left(\nu_{1},\dots,\nu_{n}\right)$-metastable at $x_{1}>\dots>x_{n}>0$ if, for $q\nearrow q_{c}$, the following hold:

$\tau_{x_{1},\delta}^{-}=O\left(1\right)$. 2. 2.

$\frac{\log\left(\tau_{x_{i},\delta}^{+}-\tau_{x_{i},\delta}^{-}\right)}{\log\left(q_{c}-q\right)}\xrightarrow{q\nearrow q_{c}}-1+\frac{1}{2\nu_{i}}$ for $i=1,\dots,n$. 3. 3.

$\tau_{x_{i+1},\delta}^{-}-\tau_{x_{i},\delta}^{+}=O\left(1\right)$ for $i=1,\dots,n$ and $x_{n+1}=0$.

We say that the critical point is $\left(\nu_{1},\dots,\nu_{n}\right)$-metastable at $x_{1}>\dots>x_{n}$ if it is $\delta$-$\left(\nu_{1},\dots,\nu_{n}\right)$-metastable at $x_{1}>\dots>x_{n}$ for small enough $\delta$. See 3.1.

The following theorem gives a full classification of the metastability properties:

Theorem 1**.**

Fix $\xi$. Then the metastable behavior is determined by one of the following cases:

1. * Case* 1.

$g_{\xi}$* attains its maximum at $1$. In this case the critical probability is $1$.* 2. * Case* 2.

$g_{\xi}$* has a unique maximum at [math]. In this case the phase transition is continuous. At the critical point*

[TABLE]

where $\nu$ is determined by the asymptotic expansion $g_{\xi}\left(x\right)=\frac{1}{q_{c}}-Cx^{\nu}+o\left(x^{\nu}\right)$. 3. * Case* 3.

The maximum of $g_{\xi}$ is attained at the points $x_{1},\dots,x_{n}$ for $1>x_{1}>\dots>x_{n}>0$, and possibly also at [math]. In this case the phase transition is discontinuous. For $i=1,\dots,n$ we may write around $x_{i}$

[TABLE]

with some $C_{i}>0$.

Then the critical point is $\left(\nu_{1},\dots,\nu_{n}\right)$-metastable at $x_{1}>\dots>x_{n}$.

Remark 1*.*

In the first case, where the critical probability is $1$, it is not clear whether or not an asymptotic expansion exists, since $g_{\xi}$ is not guaranteed to be analytic. When it does exist, one can recover a result similar to Case 3.

Finally, we show the main result – that the different metastability behaviors described above are possible:

Theorem 2**.**

Let $\nu\in\mathbb{N}$. Then there exists $\xi$ such that the phase transition is continuous, and satisfies (LABEL:criticalbehaviorcontinuous) at criticality. 2. 2.

Let $\left(\nu_{1},\dots,\nu_{n}\right)\in\mathbb{N}^{n}$. Then there exist $\xi$ and $x_{1}>\dots>x_{n}$ such that the critical point is $\left(\nu_{1},\dots,\nu_{n}\right)$-metastable at $x_{1}>\dots>x_{n}$.

4 Proofs

Proof of 1.

The idea of the proof is to notice that the main contribution to the prevalence comes from the sites close to the boundary, and then use their independence. Thus, we fix a width $w$, and consider

[TABLE]

First, we claim that $\rho_{R}\left(t\right)$ is approximated by $\rho_{R,w}\left(t\right)$ for large $w$. More accurately, we have $\left|B\left(R-w\right)\right|\leq 2^{-w}\left|B(R)\right|$, which also implies that the number of infected vertices in $B\left(R\right)\setminus B\left(R-w\right)$ is the same as the number of infected vertices in $B\left(R\right)$, up to a correction of order $2^{-w}|B(R)|$. Then

[TABLE]

We would now like to bound the distance between $\rho_{R,w}\left(t\right)$ and $1-\phi_{t}^{\xi}$. Let $\varepsilon>0$, and, by (LABEL:prevalenceonanulusclosetototal), take $w$ big enough such that $\left|\rho_{R}\left(t\right)-\rho_{R,w}\left(t\right)\right|<\frac{\varepsilon}{2}$ uniformly in $R$. Note that $\rho_{R,w}\left(t\right)$ is a weighted average of the $w$ random variables $\rho_{R,1}\left(t\right),\rho_{R-1,1}\left(t\right),\dots,\rho_{R-w+1,1}\left(t\right)$, and consider one of these variables, $\rho_{r,1}\left(t\right)$. This variable is the average of the random variables $\mathds{1}_{v\,\mbox{\tiny{is infected}}}$ for all vertices $v$ of distance $r$ from the root, and since these are independent Bernoulli random variables with mean $1-\phi_{t}^{\xi}$, and since there are at least $2^{R-w+1}$ such variables, we can use a large deviation estimate, yielding

[TABLE]

for a positive $c$ that only depends on $\varepsilon$ and on $\phi_{t}^{\xi}$. Since for $1-\phi_{t}^{\xi}$ to be far from $\rho_{R,w}\left(t\right)$ it must be far from at least one of the variables $\rho_{R,1}\left(t\right),\rho_{R-1,1}\left(t\right),\dots,\rho_{R-w+1,1}\left(t\right)$, we have

[TABLE]

Hence, $\rho_{R}(t)$ is $\varepsilon$-close to $1-\phi_{t}^{\xi}$ with probability larger than $1-we^{-c\,2^{R-w+1}}$, which concludes the proof by the Borel-Cantelli lemma. ∎

Before proving Theorems 1 and 2, we will need a couple of small results.

Claim 1*.*

$g_{k}$ is a polynomial of degree $k-1$, whose lowest degree monomial is of degree $k-r$.

Proof.

By equations 3.2 and 2.2

[TABLE]

therefore all monomials are of degree between $k-r$ and $k-1$. The coefficient of $x^{k-r}$ is ${k\choose r-1}\neq 0$, and the coefficient of $x^{k-1}$ is $\sum_{i=0}^{r-1}{k\choose i}\left(-1\right)^{i}$, which is also nonzero since $0<r-1<k$. This concludes the proof.∎

Claim 2*.*

$g_{r}\left(x\right),\dots,g_{m}\left(x\right),x^{m-r+1},\dots,x^{m-1}$ is a basis of the linear space of polynomials of degree smaller or equal to $m-1$.

Proof.

Denote $v_{1}\left(x\right)=g_{r}\left(x\right),\dots,v_{m-r+1}\left(x\right)=g_{m}\left(x\right),v_{m-r+2}\left(x\right)=x^{m-r+1},v_{m}\left(x\right)=x^{m-1}$. By 1, all $v$’s are of degree smaller or equal to $m-1$. Moreover, the matrix whose $\left(i,j\right)$ entry is the coefficient of $x^{j}$ in the polynomial $v_{i}$ is upper triangular, with nonzero diagonal. This shows that $\left\{v_{i}\right\}_{i=1}^{m}$ is indeed a basis. ∎

We will also use the following result from [6]:

Claim 3* (Claim 3.9 of [6]).*

For $\xi_{k}=\frac{r-1}{k\left(k-1\right)}$, $g_{\xi}\left(x\right)=1$.

We are now ready to prove Theorems 1 and 2.

Proof of 1.

First, we note that $g_{k}\left(1\right)=1$ for all $k$, so in particular the series $\sum_{k=r}^{\infty}\xi_{k}g_{k}\left(x\right)$ converges at $x=1$. By 1, the monomials of degree up to $n$ of the partial sum $\sum_{k=r}^{N}\xi_{k}g_{k}\left(x\right)$ are fixed once $N>n+r$. From these two facts we conclude that $g_{\xi}(x)$ is analytic in $\left(-1,1\right)$ and continuous at $1$. Thus, cases 1, 2 and 3 exhaust all possibilities.

The result will then follow from general arguments of dynamical systems near a bifurcation point. Since the exact calculations are a bit tedious, we only give here a short sketch of the argument, referring to the appendix for the complete proof.

For case 2, the expression

[TABLE]

could be estimated by comparing to the differential equation

[TABLE]

This equation could be solved explicitly, yielding the asymptotics of (LABEL:criticalbehaviorcontinuous).

For case 3, the approximate differential equation will be

[TABLE]

The solution of this equation has a plateau around $x_{i}$, whose length diverges as $(q_{c}-q)^{-1+\frac{1}{2\nu_{i}}}$. ∎

Proof of 2.

For the first part, it will be enough to show that there exist an offspring distribution $\xi$ and a polynomial $Q(x)=b_{0}+\dots+b_{r-2}x^{r-2}$ such that

$g_{\xi}\left(x\right)=\text{Const}-x^{\nu}Q(x)$. 2. 2.

$Q(x)>0$ for all $x\in\left[0,1\right]$.

This $\xi$, according to 1 and the fact that $b_{0}>0$, will indeed satisfy (LABEL:criticalbehaviorcontinuous). Rather than $\xi$, it will be easier to find a sequence $\left\{\chi_{k}\right\}_{k=r}^{\infty}$ with a finite sum together with a polynomial $P(x)=a_{0}+\dots+a_{r-2}x^{r-2}$, such that

$g_{\chi}\left(x\right)=\sum_{k}\chi_{k}g_{k}\left(x\right)=1-x^{\nu}P\left(x\right)$. 2. 2.

$\chi_{k}\geq 0$. 3. 3.

$P\left(x\right)>0$ for all $x\in\left[0,1\right]$.

Taking $\xi=\frac{1}{\sum\chi_{k}}\chi_{k}$ will then conclude the proof. Let

[TABLE]

Using 3, we may write

[TABLE]

By 1 $g_{\chi}$ is a polynomial of degree $\nu+r-2$, therefore $\sum_{k=\nu+r}^{\infty}\frac{r-1}{k\left(k-1\right)}g_{k}\left(x\right)$ equals a polynomial of degree $\nu+r-2$. Using again 1, we can define the polynomial

[TABLE]

It is left to show that $P\left(x\right)>0$ for all $x\in\left[0,1\right]$. By equations 3.2 and 2.2, $P(x)$ is non-negative and could only vanish at $x=0$. But by 1, $P(0)=\frac{r-1}{(\nu+r)(\nu+r-1)}\left(\frac{g_{\nu+r}(x)}{x^{\nu}}\right)_{x=0}\neq 0$. This concludes the first part.

Remark 2*.*

Note that, by 2, we can define the projection Pr from the space of polynomials of degree at most $r+\nu-2$ to its subspace spanned by $x^{\nu},\dots,x^{\nu+r-2}$ with kernel spanned by $g_{r}(x),\dots,g_{\nu+r-1}(x)$. Define also $M_{0}$ to be the map from the space of polynomials of degree at most $r-2$ to the space of polynomials of degree at most $r+\nu-2$ given by the multiplication by $x^{\nu}$. Then the first of the conditions above can be written as

[TABLE]

Since $\mbox{Pr}\circ M_{0}$ is bijective, this equation has a unique solution; and what we have shown in the proof is that this solution satisfies the necessary positivity conditions.

We will now prove the second part of the theorem. In analogy with the first one, we will find $\overline{\xi}$, $\overline{Q}(x)=\overline{b}_{0}+\dots+\overline{b}_{r-2}x^{r-2}$ and $x_{1}>\dots>x_{n}$ such that:

$g_{\overline{\xi}}\left(x\right)=\overline{\text{Const}}-\left(x-x_{1}\right)^{2\nu_{1}}\dots\,\left(x-x_{n}\right)^{2\nu_{n}}\overline{Q}\left(x\right)$. 2. 2.

$\overline{Q}(x)>0$ for all $x\in\left[0,1\right]$.

Similarly to the previous part, we will look for $\left\{\overline{\chi}_{k}\right\}_{k=r}^{\nu+r-1}$ and $\overline{P}(x)=\overline{a}_{0}+\dots+\overline{a}_{r-2}x^{r-2}$ satisfying:

$g_{\overline{\chi}}\left(x\right)=\sum_{k}\overline{\chi}_{k}g_{k}\left(x\right)=1-\left(x-x_{1}\right)^{2\nu_{1}}\dots\,\left(x-x_{n}\right)^{2\nu_{n}}\overline{P}\left(x\right)$. 2. 2.

$\overline{\chi}_{k}>0$. 3. 3.

$\overline{P}\left(x\right)>0$ for all $x\in\left[0,1\right]$.

Note that choosing $\nu=2\nu_{1}+\dots+2\nu_{n}$, $\chi_{k}$ (defined in (LABEL:chidef)) is strictly positive for $r\leq k\leq\nu+r-1$. Since $P$ was required to be strictly positive, we may hope that also after adding a small perturbation $\left(x_{1},\dots,x_{n}\right)$ around [math] there still exists a positive solution $\overline{P}$. More precisely, let us denote by $M_{x_{1},\dots,x_{n}}$ the multiplication by $(x-x_{1})^{2\nu_{1}}\dots\,(x-x_{n})^{2\nu_{n}}$, acting on the polynomials of degree at most $r-2$. In particular, for $x_{1},\dots,x_{n}=0$ this is $M_{0}$ defined in 2. Then, we want to show that the solution of

[TABLE]

satisfies the positivity conditions 2 and 3. By continuity of the determinant, when $(x_{1},\dots,x_{n})$ is in a small neighborhood of [math] the operator $\mbox{Pr}M_{x_{1},\dots,x_{n}}$ is invertible. Moreover, in an even smaller neighborhood of [math] the polynomial $\left(\mbox{Pr}M_{x_{1},\dots,x_{n}}\right)^{-1}\mbox{Pr}1$ will satisfy the positivity condition 3 – matrix inversion is continuous, and the set of polynomials satisfying this condition is open and contains $\left(\mbox{Pr}M_{0}\right)^{-1}\mbox{Pr}1$ by the first part of the proof. Finally, since coordinate projections of $1-(x-x_{1})^{2\nu_{1}}\dots\,(x-x_{n})^{2\nu_{n}}\left(\mbox{Pr}M_{x_{1},\dots,x_{n}}\right)^{-1}\mbox{Pr}1$ with respect to the basis defined in 2 are continuous in $(x_{1},\dots,x_{n})$, and since for $(x_{1},\dots,x_{n})=0$ condition 2 is satisfied, by taking $(x_{1},\dots,x_{n})$ in a further smaller neighborhood of [math] we are guaranteed to find a polynomial $\overline{P}$ satisfying the required conditions. ∎

5 Remarks on Two Other Phase Transitions

5.1 More Discontinuities of $\phi_{t}$

Consider, for example, $r=2$ and $\xi_{k}=\frac{3}{5}\mathds{1}_{k=2}+\frac{2}{5}\mathds{1}_{k=5}$. The function $g_{\xi}\left(x\right)$ is maximal at $g_{\xi}\left(0\right)=\frac{6}{5}$, then it has a local minimum, followed by a local maximum (see 5.1).

In this case, recalling Fact 1, $\phi_{t}^{\xi}$ will have a discontinuity at this local maximum, that is, a second phase transition occurs. We may then expect that one can find $\xi$ giving rise to as many (decreasing) local maxima of $g_{\xi}$ as we wish:

Conjecture 1**.**

Let $\nu_{1}^{\left(1\right)},\dots,\nu_{n_{1}}^{\left(1\right)},\nu_{1}^{\left(2\right)},\dots,\nu_{n_{2}}^{\left(2\right)},\dots,\nu_{n_{m}}^{\left(m\right)}$. Then there exists $g_{\xi}$, $\left\{q_{i}\right\}_{i=1}^{m},\left\{x_{j}^{\left(i\right)}\right\}_{1\leq i\leq m,\,1\leq j\leq n_{i}}$ such that $q_{i}$ is a critical point which is $\left(\nu_{1}^{\left(i\right)},\dots,\nu_{n_{i}}^{\left(i\right)}\right)$-metastable at $x_{1}^{\left(i\right)},\dots,x_{n_{i}}^{\left(i\right)}$.

5.2 Percolation of Infection

Another possible phase transition, studied in [8] for the case of regular trees, is when infinite infected clusters start to appear, but the prevalence is still smaller than $1$. Following the proof of Proposition 3.9 in [8], one sees that it applies also for the bootstrap percolation on GW trees, showing that the critical probability $q_{c}^{\left(\infty\right)}$ above which infinite clusters no longer appear is strictly bigger than $q_{c}$ defined in (LABEL:qcdef), unless $\xi_{k}=\mathds{1}_{r}$.

6 More Questions

The problem of bootstrap percolation in disordered systems raises many questions. Related to the work presented here, one may be interested in the metastable regime for other systems, such as $G_{n,p}$ or the random regular graph. Another natural problem is the analysis of the bootstrap percolation on the random graph with a given degree sequence, that has a GW local structure, with analogy to the regular tree structure of the random regular graph.

Acknowledgments

I would like to thank Cristina Toninelli for the introduction of the subject and for useful discussion, and to Lucas Benigni for his help throughout the writing of this paper.

Appendix

This paper concerns with the analysis of a phase transition originating in the appearance of a new fixed point for a certain recurrence relation, i.e., a bifurcation. In this appendix, we will try to understand in a more general context the time scaling in systems of that type. Let us then consider a sequence of reals $\left\{x_{n}\right\}_{n=0}^{\infty}$ , defined by the value $x_{0}$ and a recursion formula for $n>0$:

[TABLE]

We will also fix now some positive $\delta<1$, that will be used throughout this appendix as the window around the new fixed point in which we are interested.

First, we will study the time scaling at the bifurcation point, when the new fixed point is first created. In this case, we may expect $f$ to be tangent to the identity function at the fixed point, so we will start our discussion with the following assumptions:

Assumption A1**.**

$f$* has a fixed point $y_{0}$, such that for $y\in\left(y_{0},y_{0}+\delta\right)$:*

[TABLE]

for some $\alpha>1$, $0<\overline{c}\leq\underline{c}<\delta^{-\left(\alpha-1\right)}$.

Assumption A2**.**

$x_{0}\in\left(y_{0},y_{0}+\delta\right)$.

We first mention the following fact:

Fact A1**.**

The sequence is decreasing and bounded from below by $y_{0}$.

Proof.

By Assumption A1, $x_{n+1}<x_{n}$ whenever $x_{n}\in\left(y_{0},y_{0}+\delta\right)$. Moreover,

[TABLE]

Therefore, since $x_{0}\in\left(y_{0},y_{0}+\delta\right)$ by assumption A2, the entire sequence is in the interval $\left(y_{0},y_{0}+\delta\right)$, and it is decreasing. ∎

The following theorem will describe the asymptotic of the sequence:

Theorem A1**.**

Let $\{x_{n}\}_{n=0}^{\infty}$ be the sequence defined in (LABEL:appendixrecursion), satisfying Assumptions A1 and A2. Then

[TABLE]

where $\underline{a}=\left[\left(\alpha-1\right)\left(1-\delta\right)^{-\alpha}\underline{c}\right]^{-\frac{1}{\alpha-1}}$, $\overline{a}=\left[(\alpha-1)\overline{c}\right]^{-\frac{1}{\alpha-1}}$, and $n_{0}=\frac{(x_{0}-y_{0})^{1-\alpha}}{\left(\alpha-1\right)\left(1-\delta\right)^{-\alpha}\underline{c}}$ are all positive constants.

Proof.

Let us first define a sequence $t_{n}=\left(x_{n}-y_{0}\right)^{1-\alpha}$, and note that $t_{n}$ is positive for all $n$. Then using Fact A1 and Assumption A1, fixing $\underline{c}^{\prime}=\left(\alpha-1\right)\left(1-\delta\right)^{-\alpha}\underline{c}$ and $\overline{c}^{\prime}=\left(\alpha-1\right)\overline{c}$, we can estimate:

[TABLE]

We have used here the fact that, for any $0<z<\delta<1$, we can approximate $\left(1-z\right)^{1-\alpha}$ using its derivatives at [math] and at $\delta$:

[TABLE]

We then also use $\underline{c}t_{n-1}^{-1}=\left(x_{n}-y_{0}\right)^{\alpha-1}<\delta^{\alpha-1}<\delta$.

Finally,

[TABLE]

∎

Next, we will be interested in the behavior near the bifurcation point, just before the new fixed point appears. For this purpose we will consider a family $\left\{x_{n}^{\varepsilon}\right\}_{n=0}^{\infty}$ of sequences, each defined by the value $x_{0}^{\varepsilon}$ and a recursion formula for $n>0$:

[TABLE]

and assume:

Assumption A3**.**

There is a point $y_{0}$ such that for $\left|y-y_{0}\right|<\delta$ and $\varepsilon<\varepsilon_{0}$

[TABLE]

for an integer $\alpha>1$ and positive constants $\overline{c}$ and $\underline{c}$.

Assumption A4**.**

$0<x_{0}-y_{0}<\delta$.

In order to study the asymptotic behavior of $x^{\varepsilon}_{n}$ for small values of $\varepsilon$, we will need the following definition:

Definition A1**.**

The exit time $N_{\delta}(\varepsilon)$ is the minimal $n$ such that $x^{\varepsilon}_{n}<y_{0}-\delta$.

Replacing Fact A1 will be the following:

Fact A2**.**

For all $\varepsilon<\varepsilon_{0}$, $N_{\delta}(\varepsilon)$ is finite, and for $n<N_{\delta}(\varepsilon)$ the sequence $x^{\varepsilon}_{n}$ is decreasing.

Proof.

By Assumption A3, for $n<N_{\delta}(\varepsilon)$, if $x^{\varepsilon}_{n}<y_{0}+\delta$ then $x^{\varepsilon}_{n+1}<x^{\varepsilon}_{n}<y_{0}+\delta$. Hence, the sequence remains in the interval $(y_{0}-\delta,y_{0}+\delta)$ an long as $n<N_{\delta}(\varepsilon)$. Since in this interval the sequence is decreasing, the result follows by Assumption A4. ∎

For our analysis, we will compare this sequence to the solution of the following differential equations, that will approximate $x^{\varepsilon}_{n}-y_{0}$:

[TABLE]

The solution $\overline{\zeta}$ is strictly decreasing, and in particular one can define its inverse $\overline{t}:\,\left[-\infty,z^{\varepsilon}_{0}\right]\rightarrow\left[0,\infty\right]$, and $\overline{\tau}_{n}=\overline{t}\left(x^{\varepsilon}_{n}-y_{0}\right)$. $\underline{t}$ and $\underline{\tau}_{n}$ will be defined in the same manner. Note that these all depend on $\varepsilon$, even though this dependence is omitted from the notation. The next lemma will show that the continuous crossing times $\overline{\tau}_{n}$ and $\underline{\tau}_{n}$ are close to the discrete one, namely $n$.

Lemma A2**.**

For all $n\leq N_{\delta}(\varepsilon)$,

[TABLE]

where for all $c>0$, $\kappa_{c,\delta,\varepsilon_{0}}=\max(C_{4}\varepsilon^{2\alpha-1},2\alpha\delta^{2\alpha-1})$. $C_{4}$ is a positive constant depending on $\delta,c$ and $\varepsilon_{0}$ given explicitly in the proof, and bounded when $\delta$ and $\varepsilon_{0}$ are not too big. For example, if $\varepsilon_{0}<1$ and $c\delta^{2\alpha-1}<\frac{1}{2}$, $C_{4}<(3+4^{\alpha}c)^{4\alpha}$.

Proof.

Let $z_{n}=x_{n}-y_{0}$. Then

[TABLE]

In order to study the error term, we will use the following estimation:

Claim 4*.*

Fix $w_{0}\in(-\delta,\delta)$, and $c>0$. Let

[TABLE]

Then

[TABLE]

Proof.

We will first consider the case in which the integration interval passes through [math], that is $0<w_{0}<cw_{0}^{2\alpha}+\varepsilon$. In this case,

[TABLE]

for $C_{1}=\left(1-c\delta^{2\alpha-1}\right)^{-1}$ and $C_{2}=c\left(1-c\delta^{2\alpha-1}\right)^{-2\alpha}$.

We may then bound the nominator of the integrand for all $w\in\left[w_{0}-cw_{0}^{2\alpha}-\varepsilon,w_{0}\right]$ by

[TABLE]

where $C_{3}=\left(1+C_{2}\varepsilon_{0}^{2\alpha-1}\right)^{2\alpha}+C_{1}^{2\alpha}$.

For the denominator,

[TABLE]

Putting everything together

[TABLE]

for $C_{4}=\left[1+C_{2}\varepsilon_{0}^{2\alpha-1}\right]C_{3}$.

Next, we consider the case where the integral is over a positive interval, i.e., $w_{0}\geq cw_{0}^{2\alpha}+\varepsilon$. We can bound the integrand using convexity – for all $w\in\left(w_{0}-cw_{0}^{2\alpha}-\varepsilon,w_{0}\right)$

[TABLE]

This implies that

[TABLE]

We are left with the case $w_{0}\leq-cw_{0}^{2\alpha}-\varepsilon$, which could be analyzed using the exact same argument as the previous one to obtain the result. ∎

Using the claim we can continue with our estimation, obtaining

[TABLE]

and proving the upper bound. The lower bound could be found using the exact same calculation replacing $\underline{c}$ by $\overline{c}$. The result follows since $\overline{c}\leq\underline{c}$, and thus $\overline{\tau}_{n}\leq\underline{\tau}_{n}$ by monotonicity of the integral. ∎

We are now ready to formulate the final result:

Theorem A3**.**

Fix a family of sequences (indexed by $\varepsilon$) defined in (LABEL:appendixrecursion2) satisfying Assumptions A3 and A4, and consider their exit times $N_{\delta}(\varepsilon)$ (see Definition A1). Let $\overline{I}=\int\limits_{-\infty}^{\infty}\frac{\mbox{d}u}{\overline{c}u^{2\alpha}+1}$, $\underline{I}=\int\limits_{-\infty}^{\infty}\frac{\mbox{d}u}{\underline{c}u^{2\alpha}+1}$, and $\kappa_{\delta,0}=\max(\kappa_{\overline{c},\delta,0},\kappa_{\underline{c},\delta,0})$, where $\kappa_{\overline{c},\delta,0}$ and $\kappa_{\underline{c},\delta,0}$ are the positive constants given in Lemma A2. Assume further that $\kappa_{\delta,0}<1$. Then

[TABLE]

The factor $\frac{1}{2}$ in front of $\underline{I}$ could be removed when $\varepsilon^{-\frac{1}{2\alpha}}(x^{\varepsilon}_{0}-y_{0})^{\varepsilon}\rightarrow\infty$ as $\varepsilon\rightarrow 0$ (e.g., when $x^{\varepsilon}_{0}-y_{0}$ is bounded away from [math] uniformly in $\varepsilon$).

Proof.

This theorem is a direct consequence of the fact that $\zeta$ shows an $\varepsilon^{-1+\frac{1}{2\alpha}}$ time scaling behavior. First, note that

[TABLE]

We will then be interested in finding $\overline{t}(-\delta),\underline{t}(-\delta)$:

[TABLE]

where for $\overline{t}$ one should take $c=\overline{c}$, and $c=\underline{c}$ for $\underline{t}$.

All that is left is to use Lemma A2, finding

[TABLE]

which, since the integrals defining $\overline{I}$ and $\underline{I}$ converge, concludes the proof.∎

Remark 3*.*

When $f_{\varepsilon}$ satisfies not only Assumption A3, but also

[TABLE]

we can consider $\delta_{\varepsilon}$ that goes to [math] with $\varepsilon$, e.g. $\frac{1}{\left|\log\varepsilon\right|}$, so that $\kappa_{\delta,0}$ will converge to [math] as well. In this case, we may choose $\overline{c}_{\delta}$ and $\underline{c}_{\delta}$ that converge to $c$, and thus Theorem A3 will give the limit of $\frac{N_{\delta}(\varepsilon)}{\varepsilon^{-1+\frac{1}{2\alpha}}}$, rather than just bounds on its limsup and liminf. Such a direct application of the theorem, however, forces us to choose an initial condition $x_{0}^{\varepsilon}$ that converges to $y_{0}$ as $\varepsilon$ goes to [math]. To overcome this issue, we can use the estimation above with a fixed $\delta$ until $x_{n}$ reaches $\delta_{\varepsilon}$, which happens at $n$ of order $\int\limits_{z_{0}}^{\delta_{\varepsilon}}\frac{\mbox{d}z}{-cz^{2\alpha}-\varepsilon}\ll\varepsilon^{-1+\frac{1}{2\alpha}}$. Then restart the dynamics using the estimation with $\delta_{\varepsilon}$ until reaching $-\delta_{\varepsilon}$, which takes an order $\varepsilon^{-1+\frac{1}{2\alpha}}$ of steps, and then using again the estimation for our fixed $\delta$ show that the number of steps required to reach $-\delta$ is much smaller than $\varepsilon^{-1+\frac{1}{2\alpha}}$. This would yield

[TABLE]