Metastability of the Ising model on random regular graphs at zero temperature

Abstract

We study the metastability of the ferromagnetic Ising model on a random r-regular graph in the zero temperature limit. We prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like ${\exp (\beta (r/2+ {\mathcal {O}}(\sqrt{r}))n)}$ when the inverse temperature $\beta \rightarrow \infty$ and the number of vertices n is large enough but fixed. The proof is based on the so-called pathwise approach and bounds on the isoperimetric number of random regular graphs.

Appendix: Metastable time

In this appendix, we prove a sequence of lemmas that together give the proof of Proposition 2.4. In fact, we show that the probabilities converge exponentially or even super-exponentially fast as $\beta \rightarrow \infty$.

We prove the lower bound on the hitting time of $\boxplus$ separately for the process starting in $\boxminus$ and in a metastable state $\eta$:

Lemma 5.1

If Condition (1) holds, then, for all $\varepsilon >0$, $\delta \in (0,\varepsilon )$ and sufficiently large $\beta$,

$$\begin{aligned} {\mathbb {P}}_\boxminus \left[ \tau _\boxplus >e^{\beta (\Gamma _\ell -\varepsilon )}\right] \ge 1-e^{-\beta \delta }. \end{aligned}$$

(5.1)

Proof

To prove this lemma we need to introduce some terminology from [26]. For a non-empty set of configurations ${\mathcal {A}}$ denote by $\partial ^\mathrm{ext}{\mathcal {A}}$ the external boundary of ${\mathcal {A}}$, i.e.,

$$\begin{aligned} \partial ^\mathrm{ext}{\mathcal {A}} = \left\{ \sigma ^B \notin {\mathcal {A}} : |A\triangle B|=1 \mathrm{\ for\ some\ }\sigma ^A \in {\mathcal {A}} \right\} . \end{aligned}$$

(5.2)

We define a non-trivial cycle as a connected set of configurations ${\mathcal {C}}$ such that

$$\begin{aligned} \max _{\sigma \in {\mathcal {C}}} H(\sigma ) < \min _{\eta \in \partial ^\mathrm{ext}{\mathcal {C}}} H(\eta ). \end{aligned}$$

(5.3)

The depth $D({\mathcal {C}})$ of a non-trivial cycle ${\mathcal {C}}$ is defined as

$$\begin{aligned} D({\mathcal {C}}) = \min _{\eta \in \partial ^\mathrm{ext}{\mathcal {C}}} H(\eta ) - \min _{\sigma \in {\mathcal {C}}} H(\sigma ). \end{aligned}$$

(5.4)

We consider the cycle

$$\begin{aligned} {\mathcal {C}} = \left\{ \sigma : \Phi (\boxminus ,\sigma ) - H(\boxminus ) < \Gamma _\ell \right\} . \end{aligned}$$

(5.5)

From the definition of the communication height it follows that for any $\sigma \in {\mathcal {C}}$ it holds that $H(\sigma ) < \Gamma _\ell +H(\boxminus )$ and for any configuration $\eta \in \partial ^\mathrm{ext}{\mathcal {C}}$ it holds that $\eta \notin {\mathcal {C}}$ and hence $H(\eta ) \ge \Gamma _\ell +H(\boxminus )$. Hence, ${\mathcal {C}}$ is a non-trivial cycle.

We can thus use [26, Theorem 2.17] to conclude that, for any $\sigma \in {\mathcal {C}}$, $\varepsilon >0$, $\delta \in (0,\varepsilon )$ and sufficiently large $\beta$,

$$\begin{aligned} {\mathbb {P}}_\sigma \left[ \tau _{\partial ^\mathrm{ext}{\mathcal {C}}}>e^{\beta (D-\varepsilon )}\right] \ge 1-e^{-\beta \delta }, \end{aligned}$$

(5.6)

where

$$\begin{aligned} D= D({\mathcal {C}}) = \min _{\eta \in \partial ^\mathrm{ext}{\mathcal {C}}} H(\eta ) - \min _{\sigma \in {\mathcal {C}}} H(\sigma ) \ge \Gamma _\ell +H(\boxminus )-H(\boxminus )=\Gamma _\ell . \end{aligned}$$

(5.7)

Clearly, $\boxminus \in {\mathcal {C}}$, but $\boxplus \notin {\mathcal {C}}$ by Condition (1). Hence if the process starts from $\boxminus$ then $\tau _\boxplus \ge \tau _{\partial ^\mathrm{ext}{\mathcal {C}}}$. Hence,

$$\begin{aligned} {\mathbb {P}}_\boxminus \left[ \tau _{\boxplus }>e^{\beta (\Gamma _\ell -\varepsilon )}\right]\ge & {} {\mathbb {P}}_\boxminus \left[ \tau _{\partial ^\mathrm{ext}{\mathcal {C}}}>e^{\beta (\Gamma _\ell -\varepsilon )}\right] \nonumber \\\ge & {} {\mathbb {P}}_\boxminus \left[ \tau _{\partial ^\mathrm{ext}{\mathcal {C}}}>e^{\beta (D-\varepsilon )}\right] \ge 1-e^{-\beta \delta }. \end{aligned}$$

(5.8)

$\square$

The lower bound on the metastable time for the process starting from a metastable state is given in the next lemma:

Lemma 5.2

If $\eta$ is a metastable configuration and Condition (1) holds, then

$$\begin{aligned} \Gamma = V_\eta \ge \Gamma _\ell , \end{aligned}$$

(5.9)

and for all $\varepsilon >0$, there exists a constant K such that for all sufficiently large $\beta$

$$\begin{aligned} {\mathbb {P}}_\eta \left[ \tau _\boxplus >e^{\beta (\Gamma _\ell -\varepsilon )}\right] \ge 1-e^{-K\beta }. \end{aligned}$$

(5.10)

Proof

We prove this lemma using the same reasoning as in the proof of [14, Theorem 2.4]. Suppose that Condition (1) holds, and assume that $\Gamma <\Gamma _\ell$. Then it holds that $V_\sigma <\Gamma _\ell$ for all configurations $\sigma \ne \boxplus$. Hence, if we start with configuration $\sigma _0=\boxminus$, we can find a state $\sigma _1$ such that $H(\sigma _1)<H(\sigma _0)$ and $\Phi (\sigma _0,\sigma _1)-H(\sigma _0)<\Gamma _\ell$, i.e., there exists a path $\omega _0$ from $\sigma _0$ to $\sigma _1$ such that

$$\begin{aligned} \max _{\sigma '\in \omega _0} H(\sigma ') <\Gamma _\ell + H(\sigma _0). \end{aligned}$$

(5.11)

As long as the new configuration with lower energy is not equal to $\boxplus$, we can repeat this argument and find configurations $\sigma _2, \sigma _3, \ldots , \sigma _n$ such that $H(\sigma _0)>H(\sigma _1)>H(\sigma _2)>\cdots >H(\sigma _n)$ and $\Phi (\sigma _i,\sigma _{i+1})-H(\sigma _i)<\Gamma _\ell$ for all $i=0,2,\ldots ,n-1$, i.e, there exist paths $\omega _i$ so that

$$\begin{aligned} \max _{\sigma '\in \omega _i} H(\sigma ') <\Gamma _\ell + H(\sigma _i). \end{aligned}$$

(5.12)

Since the number of configurations is finite and the energy is strictly decreasing every step, if we choose n large enough we will end in the configuration $\sigma _n=\boxplus$. If we let $\omega$ be the concatenation of the paths $\omega _0,\omega _1,\ldots ,\omega _n$, then $\omega$ is a path from $\boxminus$ to $\boxplus$ and

$$\begin{aligned} \max _{\sigma '\in \omega } H(\sigma ') = \max _{0\le i<n} \max _{\sigma '\in \omega _i} H(\sigma ')<\Gamma _\ell + \max _{0\le i<n}H(\sigma _i) = \Gamma _\ell + H(\boxminus ). \end{aligned}$$

(5.13)

Hence,

$$\begin{aligned} \Phi (\boxminus ,\boxplus ) - H(\boxminus ) < \Gamma _\ell , \end{aligned}$$

(5.14)

which is in contradiction with Condition (1).

So, if Condition (1) holds, then $\Gamma \ge \Gamma _\ell$. Since, $\eta$ is assumed to be a metastable state $V_\eta =\Gamma \ge \Gamma _\ell$. The second statement of the lemma now immediately follows from [26, Theorem 4.1].$\square$

The upper bound on the metastable time is proved in the next lemma. Here, a function $f(\beta )$ is called super-exponentially small (SES), denoted by $f(\beta )=\mathrm{SES}$, if

$$\begin{aligned} \lim _{\beta \rightarrow \infty } \frac{1}{\beta }\log f(\beta )=-\infty . \end{aligned}$$

(5.15)

Lemma 5.3

(Upper bound on the metastable time) If Conditions (2) and (3a) hold, then, for all $\varepsilon >0$,

$$\begin{aligned} \sup _{\sigma } {\mathbb {P}}_\sigma \left[ \tau _\boxplus >e^{\beta (\Gamma _u+\varepsilon )}\right] = \mathrm{SES}. \end{aligned}$$

(5.16)

Proof

Let

$$\begin{aligned} K_V=\{\sigma : V_\sigma > V\}, \end{aligned}$$

(5.17)

be the so-called metastable set at level V. If we set $V=\Gamma _u$, then it follows from Conditions (2) and (3a) that $V_\sigma \le V$ for all $\sigma \ne \boxplus$. Hence, $K_{\Gamma _u} = \{\boxplus \}$, since $V_\boxplus =\infty$. It thus follows from [26, Theorem 3.1] that, for all $\varepsilon >0$,

$$\begin{aligned} \sup _{\sigma }{\mathbb {P}}_\sigma \left[ \tau _\boxplus >e^{\beta (\Gamma _u+\varepsilon )}\right] = \textsc {SES}. \end{aligned}$$

(5.18)

$\square$

Because of the supremum, the same bound on the hitting time of $\boxplus$ holds for the process starting from any state $\sigma$ and in particular for the process starting in $\boxminus$ and in any metastable state $\eta$.

We finally characterize when $\boxminus$ is the unique metastable state:

Lemma 5.4

If Conditions (1) and (3b) hold, then $\boxminus$ is the unique metastable state.

Proof

We want to prove that $V_\boxminus \ge \Gamma _\ell$, since then

$$\begin{aligned} \{\boxminus \} = {{\mathrm{arg\,max}}}_{\sigma \ne \boxplus } V_\sigma . \end{aligned}$$

(5.19)

Suppose that on the contrary $V_\boxminus < \Gamma _\ell$. From Condition (3b) it follows that also $V_\sigma <\Gamma _\ell$ for all $\sigma \notin \{\boxminus ,\boxplus \}$. Then, by the same reasoning as in Lemma 5.2, we can find a path $\omega$ from $\boxminus$ to $\boxplus$ such that

$$\begin{aligned} \max _{\sigma '\in \omega } H(\sigma ') <\Gamma _\ell + H(\boxminus ), \end{aligned}$$

(5.20)

and hence that

$$\begin{aligned} \Phi (\boxminus ,\boxplus )- H(\boxminus ) < \Gamma _\ell , \end{aligned}$$

(5.21)

which is in contradiction with Condition (1). $\square$