Variational Description of Gibbs-Non-Gibbs Dynamical Transitions for Spin-Flip Systems with a Kac-Type Interaction

Abstract

We continue our study of Gibbs-non-Gibbs dynamical transitions. In the present paper we consider a system of Ising spins on a large discrete torus with a Kac-type interaction subject to an independent spin-flip dynamics (infinite-temperature Glauber dynamics). We show that, in accordance with the program outlined in van Enter et al. (Moscow Math. J. 10:687–711, 2010), in the thermodynamic limit Gibbs-non-Gibbs dynamical transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the empirical density conditional on their endpoint. More precisely, the time-evolved measure is non-Gibbs if and only if this set is not a singleton for some value of the endpoint. A partial description of the possible scenarios of bifurcation is given, leading to a characterization of passages from Gibbs to non-Gibbs and vice versa, with sharp transition times. Our analysis provides a conceptual step-up from our earlier work on Gibbs-non-Gibbs dynamical transitions for the Curie–Weiss model, where the mean-field interaction allowed us to focus on trajectories of the empirical magnetization rather than the empirical density.

Appendix: Proof of Proposition 1.3

1.1 Outline

In Sects. 4.2–4.4 we sketch the proof of the LDP in Proposition 1.3(i) for deterministic initial conditions (as in Proposition 1.2(ii)), and explain why it remains true for random initial conditions. We follow the line of argument in Benois, Mourragui, Orlandi, Saada and Triolo [19] rather than Comets [12], and use various results from Kipnis and Landim [20]. The strategy of the proof consists in first proving the claim for random initial conditions drawn according to $\vartheta _\kappa ^n =\otimes _{x\in \mathbb {T}_n^d} \vartheta _{\kappa }$ with $\vartheta _{\kappa }=\mathrm {BER}(\kappa )$, $\kappa \in [0,1]$ (i.e., $\vartheta _\kappa (+1)=\kappa$ and $\vartheta _\kappa (-1) = 1-\kappa$), and afterwards replacing $\vartheta _{\kappa }^n$ by $\mu ^n$ in (1.2) with the help of Varadhan’s Lemma and Bryc’s Lemma. In Sect. 4.5 we indicate how Proposition 1.3(ii) follows.

Below we will make frequent reference to formulas in [19] and [20], so our arguments are not self-contained. We begin with the following observation.

Lemma 4.1

Suppose that $\mu$ and $\nu$ are equivalent probability measures. If $P_\mu$ and $Q_\nu$ are the laws of equivalent Markov processes with starting measures $\mu$ and $\nu$, then

$$\begin{aligned} \frac{dP_\mu }{dQ_\nu }(\bar{\eta })=\frac{d\mu }{d\nu }(\eta _0) \, \frac{dP_\mu }{dQ_\mu }(\bar{\eta }) =\frac{d\mu }{d\nu }(\eta _0) \, \frac{dP_\nu }{dQ_\nu }(\bar{\eta }). \end{aligned}$$

(4.1)

The general technique to prove an LDP relies on finding a family of mean-one positive martingales that can be written as functions of the empirical density. For Markov processes this is achieved by considering the Radon-Nikodym derivative of the original dynamics w.r.t. a small perturbation of this dynamics. It is here that Lemma 4.1 comes into play: it factorizes the Radon-Nikodym derivative into a static part and a dynamic part, as in (1.25).

1.2 Upper Bound

For initial condition $\gamma \in C(\mathbb {T}^d;[-1,+1])$ and potential $V \in C^{1,0}([0,t]\times \mathbb {T}^d)$, we denote by $\mathbb {P}^{n,V}_{\vartheta ^n_{\gamma }}$ the law of the $(\gamma ,V)$-perturbed inhomogeneous Markov process starting at

$$\begin{aligned} \vartheta ^n_\gamma =\otimes _{x\in \mathbb {T}_n^d} \vartheta _{\chi ^{-1}\left( \gamma (\tfrac{x}{n})\right) }, \end{aligned}$$

(4.2)

where $\chi :[0,1] \rightarrow [-1,+1]$ is the linear map that transforms a profile taking values in $[-1,+1]$ into a profile taking values in $[0,1]$. Details about such a perturbation and its Radon-Nikodym derivative can be found in [19, Eq. (5.8)].

1. Large deviation upper bound for compact sets. Fix $\kappa \in [0,1]$. Let $\mathcal {K} \in D_{[0,t]}(\mathcal {M}(\mathbb {T}^d))$ be compact. By Lemma 4.1, we have (recall the notation introduced in Sect. 1.5)

$$\begin{aligned} \tfrac{1}{n^d} \log \mathbb {P}^n_{\vartheta _\kappa ^n}[\bar{\pi }^n \in \mathcal {K}]&= \tfrac{1}{n^d} \log \mathbb {E}^{n,V}_{\vartheta ^n_\gamma } \left[ \left( \frac{d\mathbb {P}^n_{\vartheta _\kappa ^n}}{d\mathbb {P}^{n,V}_{\vartheta ^n_\gamma }} \, \mathbb {I}_{\mathcal {K}}\right) (\bar{\pi }^n)\right] \nonumber \\&= \tfrac{1}{n^d} \log \mathbb {E}^{n,V}_{\vartheta ^n_\gamma } \left[ \left( \frac{d\vartheta _\kappa ^n}{d\vartheta ^n_\gamma } \, \frac{d\mathbb {P}^n_{\vartheta _\kappa ^n}}{d\mathbb {P}^{n,V}_{\vartheta ^n_\kappa }} \, \mathbb {I}_{\mathcal {K}}\right) (\bar{\pi }^n)\right] \nonumber \\&= \tfrac{1}{n^d} \log \mathbb {E}^{n,V}_{\vartheta ^n_\gamma } \left[ e^{-n^d h_\gamma (\pi ^n_0)+O_\gamma (n^{-1})} \,e^{-n^d \{\hat{J}_V(\bar{\pi }^n *l^{\varepsilon ,n}) + r(V,\varepsilon ,n)\}} \, \mathbb {I}_{\mathcal {K}}(\bar{\pi }^n)\right] ,\nonumber \\ \end{aligned}$$

(4.3)

where $h_\gamma$ is the analogue of [20, Eq. (1.1), Chapter 10], $\hat{J}_V$ is defined in [19, Eq. (6.8)], $\varepsilon >0$ is small, $l^{\varepsilon ,n}$ is an approximation of the identity for $\varepsilon \downarrow 0$, and $r(V,\varepsilon ,n)$ is an error term that vanishes as $n\rightarrow \infty$ for fixed $V,\varepsilon$. By letting $n \rightarrow \infty$, optimizing over $\gamma ,V,\varepsilon$ and using the mini-max lemma, we get

$$\begin{aligned} \limsup _{n \rightarrow \infty }\tfrac{1}{n^d} \log \mathbb {P}^n_{\vartheta _\kappa ^n}[\bar{\pi }^n \in \mathcal {K}]&\le \inf _{\gamma ,V,\varepsilon } \sup _{\bar{\pi }\in \mathcal {K}}\{-h_\gamma (\pi _0)-\hat{J}_V(\bar{\pi }*l^\varepsilon )\}\nonumber \\&\le \sup _{\bar{\pi }\in \mathcal {K}} \inf _{\gamma ,V,\varepsilon } \{- h_\gamma (\pi _0) -\hat{J}_V(\bar{\pi }*l^\varepsilon )\}\nonumber \\&\le -\inf _{\bar{\pi }\in \mathcal {K}} \{I_S(\pi _0) + I^t_D(\bar{\pi })\}. \end{aligned}$$

(4.4)

The last inequality uses that $\sup _\gamma h_\gamma (\pi _0)=I_S(\pi _0)$, $\sup _V \hat{J}_V (\tilde{\pi }) = I^t_D(\tilde{\pi })$, and $\sup _\varepsilon I^t_D(\bar{\pi }*l^\varepsilon ) \ge I^t_D(\bar{\pi })$ by lower semi-continuity of $I^t_D$.

2. Exponential tightness. While in [20, Sect. 4] the initial condition is drawn from equilibrium, this is immaterial. Indeed, the proof of [19, Proposition 6.1] uses the same ideas as in [20, Sect. 4] even though the initial condition is deterministic. Hence the same computations apply to our case.

1.3 Lower Bound

• (1) Large deviation lower bound for open sets Fix $\kappa \in [0,1]$. Let $\mathcal {O} \in D_{[0,t]}(\mathcal {M}(\mathbb {T}^d))$ be open. By Lemma 4.1, we have

  $$\begin{aligned} \tfrac{1}{n^d} \log \mathbb {P}^n_{\vartheta _\kappa ^n}[\bar{\pi }^n \in \mathcal {O}]&= \tfrac{1}{n^d} \log \left\{ \mathbb {E}^{n,V}_{\vartheta ^n_\gamma }\, \left[ \frac{d\mathbb {P}^n_{\vartheta _\kappa ^n}}{d\mathbb {P}^{n,V}_{\vartheta ^n_\gamma }} (\bar{\pi }^n)~\Bigg |~ \bar{\pi }^n \in \mathcal {O}\right] \mathbb {P}^{n,V}_{\vartheta ^n_\gamma }(\mathcal {O})\right\} \nonumber \\&\ge \mathbb {E}^{n,V}_{\vartheta ^n_\gamma } \left[ \tfrac{1}{n^d} \log \frac{d\mathbb {P}^n_{\vartheta _\kappa ^n}}{d\mathbb {P}^{n,V}_{\vartheta ^n_\gamma }}(\bar{\pi }^n) ~\Bigg |~ \bar{\pi }^n \in \mathcal {O}\right] + \tfrac{1}{n^d} \log \mathbb {P}^{n,V}_{\vartheta ^n_\gamma }(\mathcal {O}),\nonumber \\ \end{aligned}$$

  (4.5)

  where we use Jensen’s inequality. By the law of large numbers for $\mathbb {P}^{n,V}_{\vartheta ^n_\gamma }$, we have

  $$\begin{aligned} w-\lim _{n\rightarrow \infty } \mathbb {P}^{n,V}_{\vartheta ^n_\gamma } = \delta _{\bar{\pi }^{\gamma , V}}, \end{aligned}$$

  (4.6)

  where $\bar{\pi }^{\gamma ,V}$ is the solution of [19, Eq. (5.5)] with initial condition $\gamma$ and potential $V$. (The proof of (4.6) follows in the same fashion as in [19]: all that is needed is that the laws of the random initial conditions converge to a law associated with continuous profile.) Hence, if $\bar{\pi }^{\gamma ,V} \in \mathcal {O}$, then $\lim _{n\rightarrow \infty } \mathbb {P}^{n,V}_{\vartheta ^n_\gamma }(\mathcal {O}) =1$. After some calculations with the Radon-Nikodym derivative, we get

  $$\begin{aligned} \liminf _{n \rightarrow \infty } \tfrac{1}{n^d} \log \mathbb {P}^n_{\vartheta _\kappa ^n}[\bar{\pi }^n \in \mathcal {O}] \ge -I^t(\bar{\pi }^{\gamma ,V}) \end{aligned}$$

  (4.7)

  with $I^t = I_S + I^t_D$.

• (2) Density arguments It remains to show that

  $$\begin{aligned} \inf _{\begin{array}{c} \gamma ,V \\ \bar{\pi }^{\gamma ,V} \in \mathcal {O} \end{array}} I^t(\bar{\pi }^{\gamma ,V}) = \inf _{\bar{\pi } \in \mathcal {O}} I^t(\bar{\pi }). \end{aligned}$$

  (4.8)

  In other words, $(\bar{\pi }^{\gamma ,V})_{\gamma , V}$ is dense with respect to $(\varrho ^w_t, I)$, i.e.,

  $$\begin{aligned} \begin{aligned}&\forall \,\bar{\pi }\in D_{[0,t]}(\mathcal {M}(\mathbb {T}^d)):\, I(\bar{\pi })< \infty , \\&\exists \,(\bar{\pi }^{\gamma _n, V_n})_{n\in \mathbb N}:\, \lim _{n\rightarrow \infty } \varrho ^w_t(\bar{\pi }^{\gamma _n, V_n},\bar{\pi }) = 0, \, \lim _{n\rightarrow \infty } I(\bar{\pi }^{\gamma _n, V_n}) = I(\bar{\pi }), \end{aligned} \end{aligned}$$

  (4.9)

  where $\varrho ^w_t$ is the supremum distance in $[0,t]$ when the marginal distance is $\varrho ^w$ (any metric that metrizes the weak topology). A density argument of this type typically exploits the fact that $I$ is lower semi-continuous and convex, but in our case $I=I^t$, which is not convex. However, in [19] density arguments are given without convexity. In order to extend these to our setting of random initial conditions, minor modifications are needed in [19, Lemma 7.5]. In particular, the space regularization of the trajectory must be done for all $s \in [0,t]$, and hence [19, Lemma 7.6] together with the arguments in [20, p. 279] prove our assertion.

1.4 Replace $\vartheta ^n_\kappa$ by $\mu ^n$

The observations made in Sects. 4.2–4.3 prove the LDP in Proposition 1.3(i), but for starting measures $\vartheta ^n_\kappa$ given by (4.2). Note that

$$\begin{aligned} \frac{d\mu ^n}{d \vartheta ^n_\kappa }= e^{n^d \beta H(\pi ^n)} \end{aligned}$$

(4.10)

with $\pi ^n \mapsto H(\pi ^n)$ in (1.6) continuous. Hence, by Lemma (4.1), Varadhan’s Lemma and Bryc’s Lemma, the LDP in Proposition 1.3(i) for starting measures $\mu ^n$ follows.

1.5 Contraction Principle

Proposition 1.3(ii) follows from Proposition 1.3(i) via the approximate contraction principle based on exponential approximation estimates. See Dembo and Zeitouni [11, Sect. 4.2].