1 Introduction↩︎

Let \(G=(V,E)\) be a finite graph and \(S\) a finite set. We study the dynamics of a class of \(S^V\)-valued spin systems in a random dynamic environment. The latter will be given by dynamical percolation, which is a \(\{0,1\}^E\)-valued Markov chain, which represents opening and closing of edges in \(E\).

Fixing \(p\in(0,1)\) and \(\lambda>0\), we define the dynamical percolation with speed \(\lambda\) and density \(p\), which we denote by \((\eta_t)_{t\geq 0}\), as follows:

• we start from an arbitrary (possibly random) initial configuration \(\eta_0\in\{0,1\}^E\);

• each edge carries an independent rate-\(\lambda\) Poisson clock;

• if the clock corresponding to edge \(e\) rings at time \(t\), we resample the value of \(\eta_t(e)\) according to \(\mathrm{Ber}(p)\), independently of anything else.

By convention, we say that the edge \(e\) is open (resp. closed) at time \(t\) if \(\eta_t(e)=1\) (resp. \(\eta_t(e)=0\)). It is easy to check that \((\eta_t)_{t\geq 0}\) is invariant with respect to Bernoulli (bond) percolation on \(G\), i.e., the measure \(\mathbb{P}_p=\mathbb{P}_p^G\), given by \[\mathbb{P}_p(\eta) ~=~ \prod_{e\in E}p^{\eta(e)}(1-p)^{1-\eta(e)}, \quad \eta\in\{0,1\}^E.\]

Consider now a collection of potentials \(\Phi=\{\Phi_{e}:e\in E\}\), where for each \(e\in E\), the map \(\Phi_e:S\times S\to\mathbb{R}\) is symmetric. That is, for \(e=xy\) and \(\sigma\in S^V\), we have \[\Phi_{xy}(\sigma(x),\sigma(y)) ~=~ \Phi_{xy}(\sigma(y),\sigma(x)).\] We define a spin system given by potentials \(\Phi\) on dynamical percolation as a stochastic process \((\sigma_t,\eta_t)_{t\geq 0}\), which evolves as follows:

• \((\eta_t)_{t\geq 0}\) is taken to be dynamical percolation on \(\{0,1\}^E\) with speed \(\lambda\) and density \(p\).

• \((\sigma_t)_{t\geq 0}\) is the \(S^V\)-valued spin component, started from arbitrary \(\sigma_0\in S^V\), which evolves in the following way:

  • each site carries an independent rate-\(1\) Poisson clock;

  • if the clock corresponding to site \(x\) rings at time \(t\), the value of \(\sigma_t(x)\) resamples according to \[\label{eq:update95conditional95probability} \mu_{\sigma_t,\eta_t}^x(\cdot) ~:=~ \frac{\exp\!\big(\!-\sum_{y:xy\in E}\eta_t(xy)\Phi_{xy}(\cdot,\sigma_t(y))\big)}{\sum_{s\in S}\exp\!\big(\!-\sum_{y:xy\in E}\eta_t(xy)\Phi_{xy}(s,\sigma_t(y))\big)}.\tag{1}\]

We will refer to this process as \(\Phi\)-spin system on dynamical percolation. The process \((\sigma_t,\eta_t)_{t\geq 0}\) is also a Markov chain; assuming that \(\Phi\) is such that the process is irreducible, we will write \(\pi\) for the corresponding invariant measure. Note, however, that \((\sigma_t)_{t\geq 0}\) by itself is not a Markov chain, as the transition rates depend on the environment. Moreover, the chain \((\sigma_t,\eta_t)_{t\geq 0}\) is not reversible, in the sense that there is no measure on \(S^V\times\{0,1\}^E\) with respect to which the chain is reversible.

We note that if in the part (i) of the definition above, we did not consider dynamical percolation but simply took \(\eta_t\equiv 1\) for all \(t\geq 0\), we would recover the Glauber dynamics for the Gibbs measure given by the Hamiltonian \(H=\sum_{xy\in E}\Phi_{xy}\). In this case we could without loss of generality simply consider only the first coordinate \((\sigma_t)_{t\geq 0}\) which would indeed be a Markov chain. Throughout the paper, we will refer to this process as the \(\Phi\)-Glauber dynamics on \(G\).

A classical example of \(\Phi\) is the one corresponding to the Ising model, where \(S=\{-1,+1\}\) and for \(xy\in E\) and \(\sigma\in S^V\) we have \[\Phi_{xy}(\sigma(x),\sigma(y)) ~=\;-\beta\sigma(x)\sigma(y),\] for some fixed \(\beta\in(0,\infty)\). In this case, if the clock corresponding to site \(x\) rings at time \(t\), we resample the value of \(\sigma_t(x)\) to \(\pm 1\) with probability proportional to \[\exp\!\Big(\!\pm\beta\sum_{y:xy\in E}\eta_t(xy)\sigma_t(y)\Big).\]

We are particularly interested in the process \((\sigma_t,\eta_t)_{t\geq 0}\) defined on a sequence of graphs \((G_N)_{N\geq 1}\), where \(G_N\) is the \(d\)-dimensional torus of size \(N\) (for some fixed \(d\geq 1\)), i.e., the vertex set is given by \(\mathbb{T}_N^d:=(\mathbb{Z}/N\mathbb{Z})^d\) and the edge set \(E(\mathbb{T}_N^d)\) is taken to be the set of nearest-neighbour edges. Note that in this case, a single collection of potentials \(\Phi\) will be replaced by a sequence \((\Phi^{(N)})_{N\geq 1}\), where \(\Phi^{(N)}=\{\Phi_{e}^{(N)}:e\in E(\mathbb{T}_N^d)\}\); we will assume no level of consistency between them a priori. Moreover, we will assume that \(p<p_c:=p_c(\mathbb{Z}^d)\), where \(p_c\) is the critical probability for bond percolation in \(\mathbb{Z}^d\). Thus, the environment typically consists of many small clusters. We will also consider the case when \(\lambda\) is small, so the environment evolves slowly.

1.0.0.1 Mixing time and main result.

Our main result concerns the mixing time of \((\sigma_t,\eta_t)_{t\geq 0}\) under the above assumptions on \(p\) and \(\lambda\), as well as some basic assumptions about the potentials. Given an irreducible aperiodic Markov chain \((X_t)_{t\geq 0}\) with stationary distribution \(\pi\), we define the mixing time \(t_\mathrm{mix}(\kappa)\) corresponding to \(\kappa\in(0,1/2)\) as \[t_\mathrm{mix}(\kappa) ~:=~ \inf\big\{t\geq 0: \max_x\|\mathrm{P}_x(X_t\in\cdot)-\pi\|_\mathrm{TV}\leq\kappa\big\},\] where \(\|\!\cdot\!\|_\mathrm{TV}\) denotes the total variation norm on measures, and \(\mathrm{P}_x\) the law of \((X_t)_{t\geq 0}\) given \(X_0=x\). For the case of classical Ising Glauber dynamics on a torus \(\mathbb{T}_N^d\), \(d\geq 2\), there exists \(\beta_c=\beta_c(d)\), so that for \(N\) large, \[\label{eq:ising95phase95transition} t_\mathrm{mix}(\kappa) ~=~ \begin{cases} \Theta(\log N), ~&\forall \beta<\beta_c,\\ \exp(\Theta(N^{d-1})), ~&\forall\beta>\beta_c; \end{cases}\tag{2}\] see for example Chapter 15 in [1]. It is natural to ask whether the same phase transition would occur if we consider the Ising model on subcritical dynamical percolation as we defined above.

Another important phenomenon related to mixing of Markov chains is that of cutoff. We say that a sequence of Markov chains with mixing times \(t_\mathrm{mix}^{(1)},t_\mathrm{mix}^{(2)},\ldots\) exhibits a cutoff if for all \(\kappa\in(0,1)\), \[\lim_{n\to\infty}\frac{t_\mathrm{mix}^{(n)}(\kappa)}{t_\mathrm{mix}^{(n)}(1-\kappa)} ~=~ 1.\] Most commonly, the sequence is taken to be a system (for example Ising Glauber dynamics) on an increasing sequence of graphs \(G_1,G_2,\ldots\), which could for example be tori of increasing size. Moreover, we say that the cutoff has a window of size \(O(w_n)\), if \(w_n=o(t_\mathrm{mix}^{(n)})\) and \[\begin{align} \lim_{\gamma\to\infty}\liminf_{n\to\infty}\max_x\|\mathrm{P}_x(X_{t_\mathrm{mix}^{(n)}-\gamma w_n}\in\cdot)-\pi\|_\mathrm{TV}~&=~ 1, \\ \lim_{\gamma\to\infty}\limsup_{n\to\infty}\max_x\|\mathrm{P}_x(X_{t_\mathrm{mix}^{(n)}+\gamma w_n}\in\cdot)-\pi\|_\mathrm{TV}~&=~ 0. \end{align}\] Despite the cutoff phenomenon being conjectured for a large class of models, it remains unproved for many classical cases. A major breakthrough in the topic was a series of papers by Lubetzky and Sly, who established in [2] the cutoff for Ising Glauber dynamics for any \(\beta<\beta_c(d)\), where \(\beta_c(d)\) is the critical value of Ising model on \(\mathbb{Z}^d\), at a window \(O(\log\log n)\), which was later improved in [3] to \(O(1)\).

Before stating our main result, we introduce some further notation. We will write \(t_\mathrm{mix}^{(N)}(\kappa)\) for the \(\kappa\)-mixing time of the \(\Phi^{(N)}\)-spin system on dynamical percolation on \(\mathbb{T}_N^d\), for some fixed \(d\geq 1\). Moreover, given a connected subgraph \(F\) of \(\mathbb{T}_N^d\), we write \(\tilde{t}_\mathrm{mix}^{(N)}(\kappa,F)\) for the \(\kappa\)-mixing time corresponding to \(\Phi^{(N)}\)-Glauber dynamics on \(F\). Now we define, for \(a>0\), \[h_\kappa^{(N)}(a) ~:=~ \sup_{F:|V(F)|\leq a} \tilde{t}_\mathrm{mix}^{(N)}(\kappa,F),\] where the supremum is over connected subgraphs of \(\mathbb{T}_N^d\); the supremum above is indeed finite as long as the chain is ergodic. Moreover, we write \[\label{eq:def95h} h_\kappa(a) ~:=~ \sup_{N\geq 1} h_\kappa^{(N)}(a).\tag{3}\]

1.0.0.2 Discussion of the proof.

The proof of the lower bound simply relies on the fact that the mixing time of \((\sigma_t,\eta_t)_{t\geq 0}\) is bounded from below by the mixing time of \((\eta_t)_{t\geq 0}\), which is known to be of order \(\lambda^{-1}\log N\), due to i.i.d. nature of the updates.

The difficult part is the upper bound, especially since many of the classic techniques (such as the spectral gap) are not readily available, due to the lack of reversibility and limited knowledge about the invariant measure \(\pi\). Instead, we use the coupling technique, which is based on constructing a coupling of two copies \((\sigma_t,\eta_t)_{t\geq 0},(\sigma_t',\eta_t')_{t\geq 0}\) of the process; the upper bound on the mixing time can be obtained by studying the probability (under this coupling) that \(\sigma_t\neq\sigma_t'\), \(\eta_t\neq\eta_t'\) for \(t\) large.

We design a coupling to exploit that the environment typically evolves more slowly than the spin component. This is achieved by first dividing the time \([0,\infty)\) into disjoint intervals of length \(\varepsilon/\lambda\) (for some small \(\varepsilon\)), which are short from the perspective of the environment but long from the perspective of the spin component, which is achieved by taking \(\lambda\ll\varepsilon\). In particular, given one such interval \([a,b)\) of length \(\varepsilon/\lambda\), the value of \(\varepsilon\) is set in a way to obtain the following:

• for any typical cluster \(\mathcal{C}\) of \(\eta_a\), which has size \(O(1)\), it is unlikely that any edge adjacent to \(\mathcal{C}\) changes its value during \([a,b)\), and in particular, the cluster remains unchanged throughout the time interval with high probability;

• the spin value at any particular site is likely to be “updated” many times during \([a,b)\).

The analysis is simplified by the fact that we can obtain a bound on mixing (see Section 2) by studying a version of the process, where we take \(\eta_t=\eta_t'\) for all \(t\geq 0\). This allows us to inspect, at the start of each time interval \([a,b)\), the disagreement between \(\sigma_a\) and \(\sigma_a'\) on sites associated to each cluster of \(\eta_a\). Due to (i), the environment in most of those clusters will remain the same during the time interval. Our coupling is constructed so that in clusters for which this is true, we obtain the following:

• if there was agreement on the sites of the cluster at time \(a\) between \(\sigma_a\) and \(\sigma_a'\), this agreement is preserved until time \(b\), and

• even if there was disagreement on at least one site of the cluster at time \(a\), there will be agreement at time \(b\) with high probability.

The main difficulty comes from clusters on which there is disagreement at time \(a\) and whose environment changes at the boundary during the time interval, allowing disagreement to propagate to other sites. We need to control how much disagreement can propagate when such bad events happen.

We solve this issue by dividing the space-time slab \(\mathbb{T}_N^d\times[0,\infty)\) into (partially overlapping) space-time boxes, of side-length \(L\) in the spatial dimension and \(C/\lambda\) in the temporal dimension, where \(L\) is a large enough value and \(C\) is large enough for the slab to contain many \(\varepsilon/\lambda\)-intervals. Furthermore, we define a notion of ajar clusters, which tell us how far a disagreement can spread from a given site within a \(\varepsilon/\lambda\)-interval in the worst-case scenario; showing that those are typically small enough is of major importance in overcoming the aforementioned issue. We then define a notion of a good (resp. bad) box, so that

• the event that a given box is good depends only on the local updates (i.e., on updates to sites and edges in the box, during the duration of the box), and

• for large enough \(t\), we are able to give a sufficient condition for agreement between \(\sigma_t\) and \(\sigma_t'\) on the entirety of the torus purely in terms of the existence of a time-oriented path of good boxes from time \(0\) to time \(t\).

The major steps are proving that a given box is good with high probability and defining good boxes carefully enough so that the above properties hold.

1.0.0.3 Related works.

Dynamical percolation was first defined in the late 90s by Häggström, Peres and Steif [4] and independently by Itai Benjamini; for a nice exposition see [5]. A model related to ours is that of a random walk on dynamical percolation, which was introduced by Peres, Stauffer and Steif [6]. Variety of results followed, for example [7]–[13]. More recently, some modifications of the model have also been explored, namely the biased random walk on dynamical percolation in [14], [15], and simple random walk on dynamical random cluster in [16], [17]. Interacting particle systems on dynamical percolation have only been considered in the past few years. The contact process on dynamical percolation was first studied by Linker and Remenik [18] and studied further in [19]–[21]. The voter model on dynamical percolation was introduced by Astoquillca [22].

There is also an existing literature studying dynamics of spin system on fixed random environments, though less related to this paper, so we will not attempt to do its span justice. As a short but diverse list of examples, one could see [23]–[26].

2 Preliminaries↩︎

In this section we give some preliminaries on the mixing times of Glauber dynamics of spin systems. We also give a simple proof of the lower bound in Theorem 1 and give a preliminary upper bound on \(t_\mathrm{mix}^{(N)}(\kappa)\), which we use in Section 3 to prove the upper bound in Theorem 1.

2.0.0.1 Total variation norm and coupling of random variables.

Let \(\mu,\nu\) be two probability measures on a measurable space \(\Omega\). The total variation distance between \(\mu\) and \(\nu\) is defined by \[\|\mu-\nu\|_\mathrm{TV}~:=~ \sup_{A\subseteq\Omega}|\mu(A)-\nu(A)|,\] where the supremum is over measurable subsets. When \(\Omega\) is finite, one more commonly employs the equivalent characterization \[\|\mu-\nu\|_\mathrm{TV}~=~ \frac{1}{2}\sum_{\omega\in\Omega}|\mu(\omega)-\nu(\omega)|.\] Recall that given two random variables \(X\) and \(Y\) defined on \(\Omega\) with respective laws \(\mu\) and \(\nu\), a coupling between \(X\) and \(Y\) is a probability measure \(\mathbf{P}\) on \(\Omega\times\Omega\), whose marginal distributions are precisely \(\mu\) and \(\nu\). It is straightforward to verify (see for example the proof of Proposition 4.2 in [27]) that given \(X,Y\) as above and any coupling \(\mathbf{P}\) between them, \[\label{eq:coupling95inequality} \|\mu-\nu\|_\mathrm{TV}~\leq~ \mathbf{P}(X\neq Y).\tag{5}\] A coupling that attains the equality, i.e., \(\|\mu-\nu\|_\mathrm{TV}=\mathbf{P}(X\neq Y)\), is called the optimal coupling. It is a classic result that such a coupling always exists on nice enough spaces – for example, it suffices to assume that \(\Omega\) is a Polish space.

2.0.0.2 Mixing times of Markov chains.

Let \((X_t)_{t\geq 0}\) be a continuous time Markov chain on a finite state space \(\Omega\) and write \(\mathrm{P}_x^{t}\) for the law of \(X_t\) conditional on \(X_0=x\). We also write \(\mathrm{P}_x\) for the law of the entire chain, conditional on \(X_0=x\), so that \(\mathrm{P}_x^t(\cdot)=\mathrm{P}_x(X_t\in\cdot)\). If \((X_t)_{t\geq 0}\) is ergodic (that is, recurrent and irreducible), then there exists a unique measure \(\pi\) that is invariant with respect to the dynamics of \((X_t)_{t\geq 0}\); moreover, the process converges weakly to \(\pi\) as \(t\to\infty\), i.e., \[X_t ~\xrightarrow{(d)}~ \pi.\] To quantify the speed of this convergence we define the notion of the mixing time: \[t_\mathrm{mix}(\kappa) ~:=~ \inf\!\Big\{t\geq0:\max_{x\in\Omega}\|\mathrm{P}_x^t-\pi\|_\mathrm{TV}\leq\kappa\Big\}, \quad\kappa\in(0,1/2).\] Classically, one writes \(t_\mathrm{mix}:=t_\mathrm{mix}(1/4)\), however this particular choice of \(\kappa\) is quite arbitrary as it turns out that \[t_\mathrm{mix}(\kappa) ~\leq~ \lceil\log_2(\kappa^{-1})\rceil t_\mathrm{mix}.\]

A classical way to approach obtaining an upper bound on \(t_\mathrm{mix}(\kappa)\) utilizes precisely the notion of coupling. Given a Markov chain generator \(\mathcal{L}\), a coupling of Markov chains with generator \(\mathcal{L}\) is a law \(\mathbf{P}\) on processes \((X_t,Y_t)_{t\geq 0}\) so that the marginal law of both \((X_t)_{t\geq 0}\) and \((Y_t)_{t\geq 0}\) is one of a Markov chain with generator \(\mathcal{L}\). In particular, letting \(\mathbf{P}_{\!x,y}\) be a coupling of Markov chains starting from \(x\) and \(y\) respectively, then \(\mathbf{P}_{\!x,y}((X_t,Y_t)\in\cdot)\) is a coupling of \(\mathrm{P}_x^t\) and \(\mathrm{P}_{y}^t\), which implies that \[\|\mathrm{P}_x^t-\mathrm{P}_y^t\|_\mathrm{TV}~\leq~ \mathbf{P}_{\!x,y}(X_t\neq Y_t).\] A simple application of the inequality \[\|\mathrm{P}_x^t-\pi\|_\mathrm{TV}~\leq~ \max_{y\in\Omega}\|\mathrm{P}_x^t-\mathrm{P}_y^t\|_\mathrm{TV}\] (see for example Lemma 4.10 in [1]) then yields that \[t_\mathrm{mix}(\kappa) ~\leq~ \inf\Big\{t\geq 0: \max_{x,y\in\Omega}\mathbf{P}_{\!x,y}(X_t\neq Y_t)\leq\kappa\Big\}.\] Note that given \(t\geq 0\) fixed, one can construct a coupling \(\mathbf{P}_{\!x,y}^t\) of \((X_t)_{t\geq 0}\) and \((Y_t)_{t\geq 0}\) started from \(x\) and \(y\), such that \[\mathbf{P}_{\!x,y}^t(X_t\neq Y_t) ~=~ \|\mathrm{P}_x^t-\mathrm{P}_y^t\|_\mathrm{TV}.\] We will call such coupling the \(t\)-optimal coupling.

2.0.0.3 Preliminary bounds on \(t_\mathrm{mix}\) and stationary environment.

We conclude this section by providing a proof of the lower bound in Theorem 1 and as well as a preliminary upper bound on \(t_\mathrm{mix}(\kappa)\) which simplifies the proof of the upper bound in Theorem 1.

Let \(\mathsf{P}_{\eta}^t\) denote the law of \(\eta_t\) conditional on \(\eta_0=\eta\). It is trivial to check (note that dynamical percolation can be understood as a random walk on a hypercube) that the mixing time of the environment \((\eta_t)_{t\geq 0}\), defined by \[t_\mathrm{mix}^\mathrm{env}(\kappa) ~:=~ \inf\Big\{t\geq 0: \max_{\eta}\|\mathsf{P}_\eta^t-\mathbb{P}_p\|_\mathrm{TV}\leq\kappa\Big\},\] where the maximum is over \(\eta\in\{0,1\}^{E(\mathbb{T}_N^d)}\), is of order \(\lambda^{-1}\log N\). Thus, the following Lemma immediately implies the lower bound in Theorem 1.

To obtain a preliminary upper bound on \(t_\mathrm{mix}(\kappa)\) one can, exploit the fact that one can construct a coupling such that after \((\eta_t)_{t\geq 0}\) and \((\eta_t')_{t\geq 0}\) coalesce, they remain in agreement. This suggests that we can assume that the environment starts from stationarity. However, as we will see below, there are some difficulties with that approach which require an additional technical result.

Given \(\sigma\in S^{\mathbb{T}_N^d}\), we write \(\mathrm{P}_\sigma^t(\cdot):=\sum_{\eta}\mathbb{P}_p(\eta)\mathrm{P}_{\sigma,\eta}^t(\cdot)\) for the law of \((\sigma_t,\eta_t)\) when started from spin configuration \(\sigma\) and from stationary environment. Moreover, given a function \(f:\{0,1\}^{E(\mathbb{T}_N^d)}\to S^{\mathbb{T}_N^d}\), write \(\mathrm{P}_f^t(\cdot):=\sum_{\eta}\mathbb{P}_p(\eta)\mathrm{P}_{f(\eta),\eta}^t(\cdot)\) for the law of chain \((\sigma_t,\eta_t)\) started from stationary environment and a spin configuration that depends on the realization of the environment via \(f\). The motivation for the latter is as follows. For fixed \(t\geq 0\) and \(f,g:\{0,1\}^{E(\mathbb{T}_N^d)}\to S^{\mathbb{T}_N^d}\), let \(\mathbf{P}_{\!f,g}^t\) be some coupling of \(\mathrm{P}_f^t\) and \(\mathrm{P}_g^t\) obtained by sampling \(\eta\sim\mathbb{P}_p\) and then running some coupling of \(\mathrm{P}_{f(\eta),\eta}^t\) and \(\mathrm{P}_{g(\eta),\eta}^t\). Then, \[\mathbf{P}_{\!f,g}^t\big((\sigma_t,\eta_t)\neq(\sigma_t',\eta_t')\big) ~\geq~ \sum_{\eta}\mathbb{P}_p(\eta)\|\mathrm{P}_{f(\eta),\eta}^t-\mathrm{P}_{g(\eta),\eta}^t\|_\mathrm{TV},\] so in particular \[\max_{f,g}\mathbf{P}_{f,g}^t\big((\sigma_t,\eta_t)\neq(\sigma_t',\eta_t')\big) ~\geq~ \sum_{\eta}\mathbb{P}_p(\eta)\max_{\sigma,\sigma'}\|\mathrm{P}_{\sigma,\eta}^t-\mathrm{P}_{\sigma',\eta}^t\|_\mathrm{TV},\] using that maximizing over \(f,g\) is equivalent to maximizing over \(\sigma,\sigma'\) for each particular realization of \(\eta\).

Having obtained this bound, the mixing result in Theorem 1 can be obtained constructing a suitable collection of couplings \(\mathbf{P}_{\!f,g}=\mathbf{P}_{\!f,g}^{N,\lambda}\) (corresponding to the system on \(\mathbb{T}_N^d\) at environment speed \(\lambda\)) as above and a sequence \(b_N(\lambda)=O(\lambda^{-1}\log N)\) with \[\inf\!\big\{t\geq 0:\max_{f,g}\mathbf{P}_{\!f,g}^{N,\lambda}(\sigma_t\neq\sigma_t')\leq\kappa/2\big\} ~\leq~ b_N(\lambda).\] Since dynamical percolation undergoes a cutoff with window \(O(1)\), we obtain the cutoff result in Theorem 1 if \(b_N(\cdot)\) is such that, if fixed \(\lambda\) is replaced by \(\lambda_N\to 0\), we also have that \(b_N(\lambda_N)=o(\lambda_N^{-1}\log N)\). For the latter to hold, it is sufficient to have \(b_N(\lambda_N)=o(\lambda_N^{-1}\log N/|\log(1-\rho_N)|)\) for some \(\rho_N\to 1\).

2.0.0.4 Some miscellaneous notation

Before moving on to proving Theorem 1, we introduce and comment on some notation and terminology that we will use throughout Section 3.

Given some percolation configuration, we shall write \(\mathcal{C}(x)\) for the connected component containing site \(x\). In particular, given dynamical percolation \((\eta_t)_{t\geq 0}\) and fixed time \(t\geq 0\), we shall write \(\mathcal{C}_t(x)\) for the connected component of \(\eta_t\) containing site \(x\).

Given a configuration \(\sigma\in S^{\mathbb{T}_N^d}\) and a subset \(\Delta\subseteq\mathbb{T}_N^d\), we will write \(\sigma(\Delta):=(\sigma(x):x\in \Delta)\in S^\Delta\) for the restriction of \(\sigma\) to \(\Delta\). Moreover, given some percolation cluster \(\mathcal{C}=(V(\mathcal{C}),E(\mathcal{C}))\), we will abuse the notation and write \(\sigma(\mathcal{C})\) for the restriction of \(\sigma\) to \(V(\mathcal{C})\). On a similar note, given \(\sigma,\sigma'\in S^{\mathbb{T}_N^d}\), we will say that \(\sigma\) and \(\sigma'\) agree on \(\mathcal{C}\) or write \(\sigma(\mathcal{C})=\sigma'(\mathcal{C})\) if \(\sigma(x)=\sigma'(x)\) for all \(x\in V(\mathcal{C})\).

Throughout the following section, we deal with a variety of different graphs and distances on them. Recall from Section 1 that we write \(d_G(\cdot,\cdot)\) for a graph distance on graph \(G\). The underlying graph for the spin system is the discrete torus \((\mathbb{T}_N^d,E(\mathbb{T}_N^d))\), whose graph distance we simply denote by \(\mathrm{dist}(\cdot,\cdot)\); we also write \[\mathrm{dist}(x,\Delta) ~:=~ \min\{\mathrm{dist}(x,y):y\in\Delta\}\] to denote the distance between a site \(x\in\mathbb{T}_N^d\) and some subset \(\Delta\subseteq\mathbb{T}_N^d\). To prove Theorem 1, we define a collection of overlapping space-time boxes, which will be indexed in a set \(\mathbb{T}_M^d\times\mathbb{N}\), where \(\mathbb{T}_M^d\) is some smaller torus indexing spatial boxes. First, by abuse of notation, we define the following the following two distances on \(\mathbb{T}_M^d\): for \(i=(i_1,\ldots,i_d)\) and \(j=(j_1,\ldots,j_d)\) on \(\mathbb{T}_M^d\), we will write \[\begin{align} \|i-j\|_1 ~&:=~ \sum_{k=1}^d d_{\mathbb{T}_M}(i_k,j_k), \\ \|i-j\|_\infty ~&:=~ \max_{1\leq k\leq d}d_{\mathbb{T}_M}(i_k,j_k), \end{align}\] where \(\mathbb{T}_M=\mathbb{Z}/M\mathbb{Z}\) is a one-dimensional torus. The reader should note that neither of the distances defined above is actually a norm, since we are on a torus. Moreover, by further abuse of notation, we define an analogue of those on the set \(\mathbb{T}_M^d\times\mathbb{N}\) via \[\begin{align} \|(i,n)-(j,m)\|_1 ~&:=~ \|i-j\|_1+|n-m|, \\ \|(i,m)-(j,m)\|_\infty ~&:=~ \max\{\|i-j\|_\infty,|n-m|\}, \end{align}\] where \(i,j\in\mathbb{T}_M^d\) and \(n,m\in\mathbb{N}\). As seen above, it will be clear whether either of the distances defined above is considered on \(\mathbb{T}_M^d\) or \(\mathbb{T}_M^d\times\mathbb{N}\) from the way we denote the entries. Towards the end of Section 3 we will also use a notion of \(\ell^p\)-paths, \(p\in\{1,\infty\}\): we say that a collection \(i_1,\ldots,i_k\in\mathbb{T}_M^d\) is an \(\ell^1\)-path (resp. \(\ell^\infty\)-path) if for any \(j=1,\ldots,k-1\), we have \(\|i_j-i_{j+1}\|_1=1\) (resp. \(\|i_j-i_{j+1}\|_\infty=1\)). An analogue definition extends to a collection \((i_1,n_1),\ldots,(i_k,n_k)\) on \(\mathbb{T}_M^d\times\mathbb{N}\).

3 Proof of Theorem 1↩︎

In this section, we complete the proof of Theorem 1. In particular, we provide the proof of the upper bound in 4 , having already obtained the lower bound via Lemma 1. In light of Lemma 2, it is sufficient to construct, for any choice of \(f\), a coupling \(\mathbf{P}_{\!f,g}\) of chains \((\sigma_t,\eta_t)_{t\geq 0}\) and \((\sigma_t',\eta_t')_{t\geq 0}\) started from \((\sigma_0,\eta_0)=(f(\eta),\eta)\) and \((\sigma_0',\eta_0')=(g(\eta),\eta)\) respectively, where \(\eta\sim\mathbb{P}_p\), for which there exist \(C_\star>0\), \(\lambda_0>0\) and \(N_0\in\mathbb{N}\), such that for all \(\lambda<\lambda_0\) and \(N>N_0\), \[\inf\!\big\{t\geq 0:\max_{f,g}\mathbf{P}_{\!f,g}\big((\sigma_t,\eta_t)\neq(\sigma_t',\eta_t')\big)\leq\kappa/2\big\} ~\leq~ \frac{C_\star}{\lambda}\log N,\] where the maximum is over functions \(f,g:\{0,1\}^{E(\mathbb{T}_N^d)}\to S^{\mathbb{T}_N^d}\).

The rest of the section proceeds as follows. First, in Section 3.1 we construct the couplings for which the above inequality will be proven. Then, in Section 3.2, we divide the space-time slab into (overlapping) boxes and define an event of a certain box being considered good (or bad). Moreover, we show that under this coupling, the probability that any particular box is good is sufficiently large, uniformly in \(f,g\). Lastly, in Section 3.3 we establish a sufficient condition for coalescence of the coupled copies of the process in terms of sufficient non-percolation of bad boxes.

3.1 Construction of the coupling↩︎

We have four processes \(\eta=(\eta_t)_{t\geq 0}\), \(\eta'=(\eta_t')_{t\geq 0}\) \(\sigma=(\sigma_t)_{t\geq 0}\) and \(\sigma'=(\sigma_t')_{t\geq 0}\). While \(\eta\) and \(\eta'\) are each independent of everything else, \(\sigma\) (resp. \(\sigma'\)) does in fact depend on \(\eta\) (resp. \(\eta'\)). The goal of defining a coupling is creating suitable dependence between \((\sigma_t,\eta_t)_{t\geq 0}\) and \((\sigma_t',\eta_t')_{t\geq 0}\). To make this precise, we define the so-called graphical representation of the process \((\sigma_t,\eta_t)_{t\geq 0}\).

Firstly, the evolution of \(\eta\) started from some starting configuration \(\eta_0\) can be constructed by (i) sampling an independent Poisson processes on \([0,\infty)\) with intensity \(\lambda\) for each edge, and (ii) decorating each mark with a value sampled from \(\mathrm{Unif}[0,1]\) distribution. This yields a collection of random triplets \((E,T,U)\in E(\mathbb{T}_N^d)\times[0,\infty)\times[0,1]\), which we denote by \(\mathrm{upd}(\eta)\). Enumerating it as \((E_i,T_i,U_i)_{i\geq 1}\) (so that \(T_i\leq T_{i+1}\) for all \(i\geq 1\)), we construct the process by starting with \(\eta_0\) and updating it at each \(T_i\), where we set the value of \(\eta_{T_i}(E_i)\) to \(1\) if \(U_i\leq p\) and to \(0\) if \(U_i>p\).

Similarly, one can construct the evolution of \(\sigma\) started from \(\sigma_0\) by (i) sampling an independent Poisson process on \([0,\infty)\) with intensity \(1\) for each site, and (ii) decorating each mark with a uniform \([0,1]\) value, which again yields a collection of triplets \((V,T,U)\in\mathbb{T}_N^d\times[0,\infty)\times[0,1]\) which we denote by \(\mathrm{upd}(\sigma)\). Enumerating it as before, we start from \(\sigma_0\) and update it at each \(T_i\), where we set the value of \(\sigma_{T_i}(V_i)\) to \(s_j\), \(j=1,\ldots,|S|\), if \[\sum_{k=1}^{j-1}\mu_{\sigma_{T_i},\eta_{T_i}}^{V_i}(s_k) ~\leq~ U ~<~ \sum_{k=1}^j\mu_{\sigma_{T_i},\eta_{T_i}}^{V_i}(s_k),\] where \(\mu_{\sigma_{T_i},\eta_{T_i}}^{V_i}\) is defined in 1 .

Given \(\sigma\), defined as above, one can construct a copy of the process \(\sigma'\) which depends on \(\sigma\), by simply making its corresponding update collection \(\mathrm{upd}(\sigma')\) a function of \(\mathrm{upd}(\sigma)\). The simplest case of that is the so-called identity coupling, where one simply lets \(\mathrm{upd}(\sigma')=\mathrm{upd}(\sigma)\). The same type of construction can be carried out for the process \(\eta\). However, this coupling alone will not help us obtain \(\sigma_t=\sigma_t'\) for some time \(t\). To achieve this, we will employ a more involved coupling.

To construct the coupling for a given, which we shall denote by \(\mathbf{Q}_{f,g}\), we start by sampling \(\eta_0\sim\mathbb{P}_p\) and \(\mathrm{upd}(\eta)\) according to the graphical representation above, and setting \(\eta_0'=\eta_0\) and \(\mathrm{upd}(\eta')=\mathrm{upd}(\eta)\). This corresponds to applying the identity coupling to both the starting configuration and the evolution of the two copies of dynamical percolation, started from stationarity. Moreover, we set \(\sigma_0=f(\eta_0)\) and \(\sigma_0'=g(\eta_0')\). We then proceed by partitioning the time axis \([0,\infty)\) into disjoint intervals \(\mathcal{I}_1,\mathcal{I}_2,\ldots\) of length \(\varepsilon/\lambda\), for some appropriate \(\varepsilon\) which is to be decided at a later point. To be precise, we define \[\mathcal{I}_k ~:=~ [t_{k-1},t_k), \quad t_k~:=~\frac{k\varepsilon}{\lambda}.\] Now we define the coupling. At each time stamp \(t_k\), \(k\geq 0\), we inspect each cluster \(\mathcal{C}\) of configuration \(\eta_{t_k}\) individually, and proceed as follows.

• If \(\sigma_{t_k}(x)=\sigma_{t_k}'(x)\) for all \(x\in V(\mathcal{C})\), then we impose identity coupling on \(\mathcal{C}\) until time \(t_{k+1}\). That is, updates of \(\sigma'\) within \(\mathcal{I}_{k+1}=[t_k,t_{k+1})\) corresponding to sites in \(\mathcal{C}\) will be given by the sample triplets \((V,T,U)\in\mathrm{upd}(\sigma)\) with \(T\in\mathcal{I}_{k+1}\) and \(V\in V(\mathcal{C})\).

• Otherwise, we consider two cases. We look at all edge updates \((E,T,U)\in\mathrm{upd}(\eta)\) with \(T\in\mathcal{I}_{k+1}\) and \(E\in E^+(\mathcal{C})\):

  • If there are none (and hence the cluster \(\mathcal{C}\) remains unchanged for entirety of \(\mathcal{I}_{k+1}\)), the process \((\sigma_t(\mathcal{C}))_{t\in\mathcal{I}_{k+1}}\) is just Glauber dynamics on a graph \(\mathcal{C}\), independent of the process outside \(\mathcal{C}\). In this case, we sample \((\sigma_t(\mathcal{C}),\sigma_t'(\mathcal{C}))_{t\in\mathcal{I}_{k+1}}\) according to the \(\varepsilon/\lambda\)-optimal coupling for \(\Phi^{(N)}\)-Glauber dynamics on \(\mathcal{C}\) given starting configurations \(\sigma_{t_k}(\mathcal{C})\) and \(\sigma_{t_k}'(\mathcal{C})\).

  • If there are edge updates to \(\mathcal{C}\) within \(\mathcal{I}_{k+1}\), then we simply run identity coupling until \(t_{k+1}\).

The verification that this indeed defines a coupling is deferred to Appendix 5, as it requires a notion defined later in this section.

One should also note that under this coupling, if \(\sigma_{t_k}(x)=\sigma_{t_k}'(x)\) for all \(x\in\mathbb{T}_N^d\) for some \(k\geq 0\), then identity coupling will be implemented on all clusters, which will result in having \(\sigma_t=\sigma_t'\) for all \(t\geq t_k\).

3.2 Renormalization: good and bad boxes↩︎

Having defined the coupling \(\mathbf{Q}_{f,g}\) in the previous section, the goal now is to study the behavior of \(\mathbf{Q}_{f,g}(\sigma_t\neq\sigma_t')\), for sufficiently large \(t\).

We renormalize the space-time \(\mathbb{T}_N^d\times[0,\infty)\) into smaller space-time boxes, whose does not depend on \(N\).

We now define the space-times boxes. Let \(L=L_\lambda\), such that \[\label{eq:L95assump} L\to\infty\quad\text{and}\quad\frac{L}{\log(1/\lambda)}\to 0, \quad\text{as}~\lambda\downarrow 0,\tag{7}\] write \(L':=8L/10\) and assume that \(L'\in\mathbb{N}\) and that \(L'\cdot M=N\) for some \(M=M_\lambda\in\mathbb{N}\). We first partition \(\mathbb{T}_N^d\) into \(M^d\) disjoint (spatial) boxes of side length \(L'\), denoted by \(\Lambda_i^\mathrm{core}\subset\mathbb{T}_N^d\), \(i\in\mathbb{T}_M^d\). More formally, given any \(i=(i_1,\ldots,i_d)\in\mathbb{T}_M^d\), we define \[\Lambda_i^\mathrm{core}~:=~ \prod_{\ell=1}^d \{i_\ell L',\ldots,(i_\ell+1)L'-1\}.\] Furthermore, define a (spatial) box \(\Lambda_i\subset\mathbb{T}_N^d\) with side length \(L\) as an \((L/10)\)-thickening³ of \(\Lambda_i^\mathrm{core}\). We will refer to \(\Lambda_i^\mathrm{core}\) as the core of \(\Lambda_i\).

Fixing a constant \(C>0\) and writing \[\mathbf{I}_n ~=~ \mathbf{I}_{n}^{\lambda,C} ~:=~ \left[\frac{(n-1)C}{\lambda},\frac{(n+1)C}{\lambda}\right), \quad n\in\mathbb{N},\] we define the space-time boxes as \[\mathsf{B}_{i,n} ~:=~ \Lambda_i\times\mathbf{I}_n, \quad i\in\mathbb{T}_M^d,n\in\mathbb{N}.\] Notice that \(\mathsf{B}_{i,n}\cap\mathsf{B}_{i',n'}=\emptyset\) iff \(\|i-i'\|_\infty> 1\) or \(|n-n'|> 2\). Moreover, the second half in the temporal dimension of \(\mathsf{B}_{i,n}\) is precisely the first half of \(\mathsf{B}_{i,n+1}\).

To cope with the fact that given \(x\in\mathbb{T}_N^d\), the cluster containing \(x\) changes with time, we introduce a more robust notion of a cluster. Given \(k\geq 1\), we say that an edge \(e\) is ajar during \(\mathcal{I}_k\), if there exists \(t\in\mathcal{I}_k\) such that \(\eta_t(e)=1\). This induces a natural configuration \(\zeta_k\in\{0,1\}^{E(\mathbb{T}_N^d)}\) given by \[\zeta_k(e) ~:=~ \mathbb{1}\{e~\text{is ajar during}~\mathcal{I}_k\}, \quad e\in E(\mathbb{T}_N^d).\] It is immediate that for each \(t\in\mathcal{I}_k\), \(\eta_t\leq\zeta_k\) in the pointwise sense, which in turn implies that for any choice of \(x\in\mathbb{T}_N^d\), \[V(\mathcal{C}_t(x)) ~\subseteq~ V(\mathcal{C}_k^\mathrm{ajar}(x)), \quad \forall t\in\mathcal{I}_k,\] where \(\mathcal{C}_k^\mathrm{ajar}\) refers to \(\zeta_k\)-clusters. Note that for any \(y\notin V(\mathcal{C}_k^\mathrm{ajar}(x))\), there is no \(t\in\mathcal{I}_k\) and \(x'\in V(\mathcal{C}_k^\mathrm{ajar}(x))\) for which \(\eta_t(x',y)=1\), so we find that the restrictions of \((\sigma_t)_{t\in[a,b]}\) to \(V(\mathcal{C}_k^\mathrm{ajar}(x))\) and \(\mathbb{T}_N^d\setminus V(\mathcal{C}_k^\mathrm{ajar}(x))\) are independent. It is not difficult to see (as demonstrated in the proof of Lemma 3 below) that by taking \(\varepsilon\) sufficiently small, the law of \(\zeta_k\) can be stochastically dominated by subcritical Bernoulli percolation, which again tells us that typically, during \(\mathcal{I}_k\), any two sites that are sufficiently distant from each other evolve independently. We use this fact to limit how far any disagreement that exists at time \(t_{k-1}\) can spread during \(\mathcal{I}_k\): note that indeed, if \(x\neq y\) are such that

• \(y\notin V(\mathcal{C}_k^\mathrm{ajar}(x))\),

• \(\sigma_{t_{k-1}}\) and \(\sigma_{t_{k-1}}'\) do not agree on \(\mathcal{C}_k^\mathrm{ajar}(x)\), and

• \(\sigma_{t_{k-1}}\) and \(\sigma_{t_{k-1}}'\) agree on \(\mathcal{C}_k^\mathrm{ajar}(y)\),

then \(\sigma_t\) and \(\sigma_t'\) agree on \(\mathcal{C}_k^\mathrm{ajar}(y)\) for all \(t\in\mathcal{I}_{k}\); in particular, their disagreement on \(\mathcal{C}_k^\mathrm{ajar}(x)\) does not affect the agreement on \(\mathcal{C}_k^\mathrm{ajar}(y)\) during \(\mathcal{I}_{k}\).

Another consequence of \(\eta_t\)-clusters typically being small is, that by selecting \(\varepsilon\) small, it is likely that no edge adjacent to some typical cluster receives an update during \(\mathcal{I}_k\). This not only yields that this cluster coincides with the associated ajar cluster, but also that the restriction of \((\sigma_t)_{t\in\mathcal{I}_{k}}\) to this cluster (which remains static during \(\mathcal{I}_k\)) is just classic Glauber dynamics on a fixed graph. By then taking \(\lambda\) sufficiently small (in order to make \(\varepsilon/\lambda\) large), we can make the probability that \(\sigma\) and \(\sigma'\) reach agreement on this cluster (under the optimal coupling) arbitrarily close to \(1\). This is central to our proof later, once we control the sizes of clusters.

We introduce some notation for subsets of \(\mathrm{upd}(\sigma)\) and \(\mathrm{upd}(\eta)\), contained in a specific space-time box. First we define the first and second half of the interval \(\mathbf{I}_n\) via \(\mathbf{I}_n^{(1)}:=[(n-1)C/\lambda,nC/\lambda)\) and \(\mathbf{I}_n^{(2)}:=[nC/\lambda,(n+1)C/\lambda)\). Considering a box \(\mathsf{B}_{i,n}\), \(i\in\mathbb{T}_M^d,n\in\mathbb{N}\), we write for \(k=1,2\), \[\begin{align} \mathrm{upd}_{i,n}^{(k)}(\sigma) ~&:=~ \{(V,T,U)\in\mathrm{upd}(\sigma):(V,T)\in\Lambda_i\times\mathbf{I}_n^{(k)}\}, \\ \mathrm{upd}_{i,n}^{(k)}(\eta) ~&:=~ \{(E,T,U)\in\mathrm{upd}(\eta):(E,T)\in E(\Lambda_i)\times\mathbf{I}_n^{(k)}\}, \end{align}\] as well as \[\begin{align} \mathrm{upd}_{i,n}(\sigma) ~&:=~ \mathrm{upd}_{i,n}^{(1)}(\sigma)\cup\mathrm{upd}_{i,n}^{(2)}(\sigma), \\ \mathrm{upd}_{i,n}(\eta) ~&:=~ \mathrm{upd}_{i,n}^{(1)}(\eta)\cup\mathrm{upd}_{i,n}^{(2)}(\eta). \end{align}\] We wish to define a notion of a box \(\mathsf{B}_{i,n}\) being “good”, which is measurable with respect to the \(\sigma\)-algebra generated by \(\mathrm{upd}_{i,n}(\sigma)\), \(\mathrm{upd}_{i,n}(\sigma')\) and \(\mathrm{upd}_{i,n}(\eta)\). Before doing so, we introduce some further notation. We write \(\overline{\Lambda_i^\mathrm{core}}\) for the \((L/100)\)-thickening of \(\Lambda_i^\mathrm{core}\). Note that following this definition, \(\Lambda_i^\mathrm{core}\subset\overline{\Lambda_i^\mathrm{core}}\subset\Lambda_i\). Moreover, we write, for \(k=1,2\),

• \(\mathcal{S}_{i,n}^{(k)}\) for the collection of locally finite subsets of \((\mathbb{T}_N^d\setminus\Lambda_i)\times\mathbf{I}_n^{(k)}\times[0,1]\), and

• \(\mathcal{E}_{i,n}^{(k)}\) for the collection of locally finite subsets of \((E(\mathbb{T}_N^d)\setminus E(\Lambda_i))\times\mathbf{I}_n^{(k)}\times[0,1]\),

as well as \[\begin{align} \mathcal{S}_{i,n} ~&:=~ \{\Xi_1\cup\Xi_2:\Xi_k\in\mathcal{S}_{i,n}^{(k)}\}, \\ \mathcal{E}_{i,n} ~&:=~ \{\Upsilon_1\cup\Upsilon_2:\Upsilon_k\in\mathcal{E}_{i,n}^{(k)}\}. \end{align}\]

Note also, that by the nature of the events defined in Definition 1, their probabilities do not depend on the particular choice of \(f,g\), and hence all the bounds will be uniform in \(f,g\). Thus, for the rest of the section, we will suppress the subscript and simply write \(\mathbf{Q}\) in place of \(\mathbf{Q}_{f,g}\).

Before proving Claim 4, we briefly provide another claim that we use will use in the proof. Let \(F\) be a connected subgraph of \(\mathbb{T}_N^d\), and let \(\mathbf{P}_{\!\mathrm{opt},F}^T\) denote the \(T\)-optimal coupling for \(\Phi^{(N)}\)-Glauber dynamics on \(F\). Letting \[\delta_F^{(N)}(T) ~:=~ \max_{\sigma_0,\sigma_0'}\mathbf{P}_{\!\mathrm{opt},F}^T(\sigma_T\neq\sigma_T'),\] where \((\sigma_t)_{t\geq 0},(\sigma_t')_{t\geq 0}\) are two copies of the process, started from \(\sigma_0,\sigma_0'\in S^{V(F)}\), we define \[\delta(a,T) ~:=~ \sup_{N\geq 1}\max_{F:|V(F)|\leq\log^2 a}\delta_F^{(N)}(T),\] where the maximum is over connected subgraphs of \(\mathbb{T}_N^d\). Given this definition, on any cluster of size at most \(\log^2 L\), where \(\varepsilon/\lambda\)-optimal coupling is ran during an interval \(\mathcal{I}_k\), the copies of the process will agree at the end of \(\mathcal{I}_k\) with probability at least \(1-\delta(L,\varepsilon/\lambda)\).

3.3 Sufficient condition for agreement↩︎

As explained above, the event that \(\mathsf{B}_{i,n}\) is good gives us several important bits of information, the first being that at \(t=(n+1)C/\lambda\), \(\sigma_t\) and \(\sigma_t'\) agree on the entirety of \(\overline{\Lambda_i^\mathrm{core}}\supset\Lambda_i^\mathrm{core}\). Following this, one could simply wait for the first \(n\), such that \(\mathsf{B}_{i,n}\) is good for all \(i\in\mathbb{T}_M^d\). This however, regardless of \(\mathbf{Q}(\mathsf{B}_{i,n}~\text{is good})\) being arbitrarily high, takes too long to happen.

The other important information is obtained from Condition([A3]). In particular, if the preceding box, \(\mathsf{B}_{i,n-1}\), is also good, then we obtain that \(\sigma_t\) and \(\sigma_t'\) agree on the entirety of \(\Lambda_i^\mathrm{core}\) for all \(t\in\mathbf{I}_n^{(2)}=[nC/\lambda,(n+1)C/\lambda)\). We will exploit this to obtain a sufficient condition for agreement of processes at times of form \(nC/\lambda\).

In what follows we will write, for a given \(n\geq 1\), we will write \(J^{(n)}=\{j\in\mathbb{T}_M^d:\mathsf{B}_{j,n}~\text{is good}\}\) and denote by \(A_1^{(n)},\ldots,A_{\ell_n}^{(n)}\) the connected components of bad boxes in the \(\ell^\infty\)-distance, i.e., the coarsest partition of \(\mathbb{T}_M^d\setminus J^{(n)}\) such that for any \(i\in A_k^{(n)}\) and \(j\in A_{k'}^{(n)}\) with \(k\neq k'\) we have \(\|i-j\|_\infty>1\). Moreover, for \(k\in\{1,\ldots,\ell_n\}\), we will write \(J_k^{(n)}\) for all \(j\in J^{(n)}\) for which there exists \(i\in A_k^{(n)}\) such that \(\|i-j\|_\infty=1\).

We can state and prove a sufficient condition for agreement.

Define now, for each \(i\in\mathbb{T}_M^d\) and \(n\in\mathbb{N}\), a random variable \[X_{i,n} ~:=~ \mathbb{1}_{\{\mathsf{B}_{i,n}~\text{is good}\}},\] and write \(\mathrm{P}_{\!\mathrm{good}}\) for the law of \((X_{i,n}:i\in\mathbb{T}_M^d,n\in\mathbb{N})\) under \(\mathbf{Q}\). Since the event \(\{\mathsf{B}_{i,n}~\text{is good}\}\) depends only on updates within \(\mathsf{B}_{i,n}\), it is independent of all the boxes which it does not intersect. Since \(\mathbf{Q}(X_{i,n}=1)\) can be made arbitrarily high, it follows from the result of Liggett, Schonmann and Stacey (Theorem 0.0 in [28]) that there exists a \(\rho=\rho(\mathbf{Q}(X_{i,n}=1))\), which too can be made arbitrarily high, such that \[\mathbb{P}_{\rho}^\mathrm{site}~\preceq_\mathcal{D}~ \mathrm{P}_{\!\mathrm{good}},\] where \(\mathbb{P}_\rho^\mathrm{site}\) denotes a Bernoulli site percolation on \(\mathbb{T}_M^d\times\mathbb{N}\) with density \(\rho\). Since the event that there exist no \(\ell^\infty\)-paths of \(0\)-sites from \(\mathbb{T}_M^d\times\{1\}\) to \(\mathbb{T}_M^d\times\{n\}\) is increasing, it is sufficient for us to give a lower bound on its probability w.r.t. \(\mathbb{P}_\rho^\mathrm{site}\), provided it is high enough to yield the result.

We are finally equipped to give the proof of the main Theorem.

4 Lower bound on the mixing time↩︎

In the previous two chapters, we carried out the proof that for sufficiently small \(\lambda>0\), we have \(t_\mathrm{mix}=\Theta(\lambda^{-1}\log N)\), by first demonstrating that \[t_\mathrm{mix}^\mathrm{env}(\kappa) ~\leq~ t_\mathrm{mix}(\kappa) ~\leq~ t_\mathrm{mix}^\mathrm{env}(\kappa/2) + \inf\{t\geq 0:\max_{f,g}\mathbf{P}_{\!f,g}(\sigma_t\neq\sigma_t')\leq\kappa/2\},\] where \(\mathbf{P}_{\!f,g}\) is any coupling as in Lemma 2. Then, we showed that \(t_\mathrm{mix}^\mathrm{env}(\kappa)=\Theta(\lambda^{-1}\log N)\) and that for an appropriate choice of coupling \(\mathbf{P}_{\!f,g}\) we have \[\inf\{t\geq 0:\max_{f,g}\mathbf{P}_{\!f,g}(\sigma_t\neq\sigma_t')\leq\kappa/2\}~=~O(\lambda^{-1}\log N).\] However, since we only provided an upper bound for the quantity above, the question remains whether one can do better asymptotically. This is a particularly important question to address, as proving that it is also \(o(\lambda^{-1}\log N)\) would imply a cutoff at the same location as the one for the random environment.

Heuristically, however, this does not seem likely to be true. For the classical case (i.e., without the random environment), it was proven in [29] that any nonredundant⁵ Glauber dynamics on a sequence of finite undirected graphs of bounded degree has mixing time of order \(\Omega(\log N)\). However, the proof in [29] is restricted to reversible Markov chains.

Below, we provide an analogue result in our setting for a large class of interactions, corresponding to monotone measures, which includes for example the Ising model. The proof is heavily inspired by the one in [29].

Consider some linear order on \(S\). We say that a measure \(\mu\) on \(S^{\mathbb{T}_N^d}\) is monotone if the following holds for \(X\sim\mu\) and any \(\xi,\zeta\in S^{\mathbb{T}_N^d\setminus\{x\}}\) with \(\xi\leq\zeta\) and \(\mu(\xi),\mu(\zeta)>0\): \[\mu\big(X(x)\geq s\big|X(\mathbb{T}_N^d\setminus\{x\})=\xi\big) ~\leq~ \mu\big(X(x)\geq s\big|X(\mathbb{T}_N^d\setminus\{x\})=\zeta\big),\] for any \(s\in S\) and \(x\in\mathbb{T}_N^d\). A very useful notion that exists in this setting is that of a monotone coupling. In the context of Glauber dynamics, that translates to an ability to couple two copies \((\sigma_t)_{t\geq 0}\) and \((\sigma_t')_{t\geq 0}\) of the process with starting configurations \(\sigma_0\leq\sigma_0'\) in such a way that \(\sigma_t\leq\sigma_t'\) for all \(t\geq 0\). We will moreover assume that the invariant measures corresponding to \(\Phi^{(N)}\)-spin systems on dynamical percolation satisfy the following non-triviality condition: \[\label{eq:nonrtiv95condition} \begin{align} &\text{there exists a partition (S^+,S^-) of S with s_+>s_- for any s_+\in S^+,s_-\in S^-,} \\ &\text{such that there exists \chi\in(0,1/2), so that} \\ &\quad\quad\quad\quad\quad\quad\quad\quad\chi ~\leq~ \pi(\sigma(x)\in S^-) ~\leq~ 1-\chi\quad, \quad\forall x\in\mathbb{T}_N^d, \\ &\text{uniformly in N.} \end{align}\tag{14}\]

In what follows, we write \(t_\mathrm{mix}^\mathrm{sta}(\cdot)\) for the mixing time for the corresponding spin system on dynamical percolation started from \(\eta_0\sim\mathbb{P}_p\).

5 Validity of coupling \(\mathbf{Q}_f\)↩︎

Here we verify that \(\mathbf{Q}_{f,g}\) in fact defines a coupling of \(\mathrm{P}_f\) and \(\mathrm{P}_g\), in the sense that for each \(t\geq 0\), the marginals of the law of \((\sigma_t,\eta_t)\) and \((\sigma_t',\eta_t')\) under \(\mathbf{Q}_{f,g}\) correspond to \(\mathrm{P}_f^t\) and \(\mathrm{P}_g^t\) respectively.

It is immediate, due to the implementation of the identity coupling that both \((\eta_t)_{t\geq 0}\) and \((\eta_t')_{t\geq 0}\) have law \(\mathsf{P}:=\sum_{\eta}\mathbb{P}_p(\eta)\mathsf{P}_\eta(\cdot)\). In particular, for any \(t\geq 0\), the marginal law of \(\eta_t\) (resp. \(\eta_t'\)) under \(\mathbf{Q}_{f,g}\) is \(\mathbb{P}_p\).

We now write \(\mathcal{I}:=[0,\varepsilon/\lambda]\) and recall that for fixed \(k\geq 0\), we defined \(t_k=k\varepsilon/\lambda\). For \(t,t'\geq 0\), we write \(\mathcal{F}_{t,t'}\) for the \(\sigma\)-algebra generated by \((V,T,U)\in\mathrm{upd}(\sigma)\cup\mathrm{upd}(\sigma')\) such that \(T\leq t\) and \((E,T,U)\in\mathrm{upd}(\eta)\cup\mathrm{upd}(\eta')\) such that \(T\leq t'\), as well as the realization of the (random) initial configurations. Additionally, we write \(\mathcal{F}_t:=\mathcal{F}_{t,t}\). It is sufficient to verify that for any given \(\xi\in S^{\mathbb{T}_N^d}\), \(k\geq 0\) and \(s\in\mathcal{I}\), \[\begin{align} \mathbf{Q}_{f,g}(\sigma_{t_k+s}=\xi|\mathcal{F}_{t_k}) ~&=~ \mathrm{P}_{\sigma_{t_k},\eta_{t_k}}^s(\xi), \tag{18} \\ \mathbf{Q}_{f,g}(\sigma_{t_k+s}'=\xi|\mathcal{F}_{t_k}) ~&=~ \mathrm{P}_{\sigma_{t_k}',\eta_{t_k}'}^s(\xi). \tag{19} \end{align}\] By definition of the coupling, i.e., since \((\sigma_{t},\eta_t)_{t\geq 0}\) is evolved using \(\mathrm{upd}(\sigma)\) and \(\mathrm{upd}(\eta)\), it is immediate that 18 holds; it remains to verify 19 . Writing \(\mathbf{E}_{f,g}\) for the expectation under \(\mathbf{Q}_{f,g}\), it follows from the tower property that \[\mathbf{Q}_{f,g}(\sigma_{t_k+s}'=\xi|\mathcal{F}_{t_k}) ~=~ \mathbf{E}_{f,g}\big[\mathbf{Q}_{f,g}(\sigma_{t_k+s}'=\xi|\mathcal{F}_{t_k,t_{k+1}})\big|\mathcal{F}_{t_k}\big];\] it is thus sufficient to verify that \[\mathbf{Q}_{f,g}(\sigma_{t_k+s}'=\xi|\mathcal{F}_{t_k,t_{k+1}}) ~=~ \mathrm{P}_{\sigma_{t_k}',\eta_{t_k}'}^s\big(\xi\big|(\eta_t')_{t\in\mathcal{I}}\big).\] Recalling the notion of ajar clusters from Section 3.2, we slightly abuse the notation to write \(\mathcal{C}_{k+1}^\mathrm{ajar}\) for the collection of ajar clusters associated with the interval \(\mathcal{I}_{k+1}=[t_k,t_{k+1})\). We note that \(\mathcal{C}_{k+1}^\mathrm{ajar}\) forms a partition of \(\mathbb{T}_N^d\) and recall that for any distinct \(\mathcal{C},\mathcal{C}'\in\mathcal{C}_{k+1}^\mathrm{ajar}\), \[\big(\sigma_{t}'(\mathcal{C}):t\in\mathcal{I}_{k+1}\big)\quad\text{and}\quad\big(\sigma_{t}'(\mathcal{C}'):t\in\mathcal{I}_{k+1}\big)\] are independent. Thus, since the collection \(\mathcal{C}_{k+1}^\mathrm{ajar}\) is measurable with respect to \(\mathcal{F}_{t_k,t_{k+1}}\), we obtain the following decomposition: \[\label{eq:ajar95decomp} \mathbf{Q}_{f,g}(\sigma_{t_k+s}'=\xi|\mathcal{F}_{t_k,t_{k+1}}) ~=~ \prod_{\mathcal{C}\in\mathcal{C}_{k+1}^\mathrm{ajar}}\mathbf{Q}_{f,g}\big(\sigma_{t_k+s}'(\mathcal{C})=\xi(\mathcal{C})\big|\mathcal{F}_{t_k,t_{k+1}}\big).\tag{20}\] Fixing some \(\mathcal{C}\in\mathcal{C}_{k+1}^\mathrm{ajar}\), we distinguish between two scenarios:

• If \(\sigma_{t_k}(\mathcal{C})\neq\sigma_{t_k}'(\mathcal{C})\) and no edge updates occurred to the edges in the \(\mathcal{C}\), i.e., there exist no \((E,T,U)\in\mathrm{upd}(\eta)\) with \(E\in E^+(\mathcal{C})\) and \(t_k\leq T<t_{k+1}\), then the restriction of \(\mathbf{Q}_{f,g}\) to \(\mathcal{C}\) and \(\mathcal{I}_{k+1}\) corresponds to the \((\varepsilon/\lambda)\)-optimal coupling on \(\mathcal{C}\) (described in Section 2), keeping in mind that the absence of edge updates implies that \(\mathcal{C}=\mathcal{C}_{s}(x)\) for arbitrary \(s\in\mathcal{I}_{k+1}\) and \(x\in V(\mathcal{C})\). In particular, \[\begin{align} \mathbf{Q}_{f,g}\big(\sigma_{t_k+s}'(\mathcal{C})=\xi(\mathcal{C})\big|\mathcal{F}_{t_k,t_{k+1}}\big) ~&=~ \mathrm{P}_{\sigma_{t_k}',\eta_{t_k}'}\big(\sigma_s'(\mathcal{C})=\xi(\mathcal{C})\big|(\eta_t')_{t\in\mathcal{I}}\big), \end{align}\] noting that \((\sigma_{t_k}',\eta_{t_k}')\) is understood as a realization of random variables, measurable with respect to \(\mathcal{F}_{t_k,t_{k+1}}\).

• In the opposite case, the restriction of \(\mathbf{Q}_{f,g}\) to \(\mathcal{C}\) and \(\mathcal{I}_{k+1}\) corresponds to the identity coupling and hence \[\begin{align} \mathbf{Q}_{f,g}\big(\sigma_{t_k+s}'(\mathcal{C})=\xi(\mathcal{C})\big|\mathcal{F}_{t_k,t_{k+1}}\big) ~&=~ \mathrm{P}_{\sigma_{t_k}',\eta_{t_k}'}\big(\sigma_s'(\mathcal{C})=\xi(\mathcal{C})\big|(\eta_t')_{t\in\mathcal{I}}\big). \end{align}\]

It thus follows that \[\begin{align} \mathbf{Q}_{f,g}(\sigma_{t_k+s}'=\xi|\mathcal{F}_{t_k,t_{k+1}}) ~&=~ \prod_{\mathcal{C}\in\mathcal{C}_{k+1}^\mathrm{ajar}} \mathrm{P}_{\sigma_{t_k}',\eta_{t_k}'}\big(\sigma_s'(\mathcal{C})=\xi(\mathcal{C})\big|(\eta_t')_{t\in\mathcal{I}}\big) \\ &=~ \mathrm{P}_{\sigma_{t_k}',\eta_{t_k}'}^s\big(\xi\big|(\eta_t')_{t\in\mathcal{I}}\big), \end{align}\] concluding the proof.

References↩︎

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1. Department of Mathematics, King’s College London. Email: a.stauffer@kcl.ac.uk.↩︎

2. Department of Mathematics, King’s College London. Email: oskar.vavtar@kcl.ac.uk.↩︎

3. The \(r\)-thickening of a set \(\Lambda\subseteq\mathbb{T}_N^d\) is defined as the set \(\{x:\mathrm{dist}_\infty(x,\Lambda)\leq r\}\). In particular, if \(\Lambda=\{-k,\ldots,k\}^d\), then its \(r\)-thickening is \(\{-(k+r),\ldots,k+r\}^d\).↩︎

4. The choice of \(2\) here is completely arbitrary, any fixed real number strictly larger \(1\) works.↩︎

5. This condition requires that under the dynamics, at least two different spin values can appear on any site.↩︎