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Random-cluster dynamics in \({{\mathrm{\mathbb {Z}}}}^2\)

Abstract

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an \(n \times n\) box in the Cartesian lattice \({{\mathrm{\mathbb {Z}}}}^2\). Our main result is a \(O(n^2\log n)\) upper bound for the mixing time at all values of the model parameter p except the critical point \(p=p_c(q)\), and for all values of the second model parameter \(q\ge 1\). We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in \({{\mathrm{\mathbb {Z}}}}^2\). It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense.

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Notes

  1. We say that an event occurs with high probability if it occurs with probability approaching 1 as \(n \rightarrow \infty \).

  2. For a pair of functions \(f,g:{{\mathrm{\mathbb {N}}}}\rightarrow {{\mathrm{\mathbb {R}}}}^+\), we say that \(f(n) = O(g(n))\) (resp., \(f(n) = \varOmega (g(n))\)) if there exists constants \(c,n_0 > 0\) such that \(f(n) \le c \cdot g(n)\) (resp., \(f(n) \ge c \cdot g(n))\) for all \(n > n_0\). We say that \(f(n) = \varTheta (g(n))\) when \(f(n) = O(g(n))\) and \(f(n) = \varOmega (g(n))\).

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Acknowledgments

The authors would like to thank Fabio Martinelli and Allan Sly for helpful suggestions. We would like also to thank the anonymous referees whose valuable comments improved the exposition of the results.

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Correspondence to Antonio Blanca.

Additional information

Antonio Blanca was supported in part by an NSF Graduate Research Fellowship and NSF Grant CCF-1420934. Alistair Sinclair was supported in part by NSF Grant CCF-1420934.

Appendix: Proof of Lemma 6.2

Appendix: Proof of Lemma 6.2

We show first that the measure that results from conditioning on the state of a single edge maintains the exponential decay of finite volume connectivities (3).

Fact 7.2

Let \(p < p_c(q)\), \(q \ge 1\), and let \(\eta \) be a boundary condition for \({{\mathrm{{\Lambda }}}}= ({{\mathrm{{\Lambda }}}}_n,E_n)\). Consider a copy \(\{Y_t\}\) of the continuous time Glauber dynamics on \({{\mathrm{{\Lambda }}}}\), and assume \(Y_0\) is sampled from the distribution \(\mu ^\eta _{{{\mathrm{{\Lambda }}}},p,q}(\,\cdot \,|\,e=b)\), for some \(e \in E_n\) and \(b \in \{0,1\}\). Then, for all \(u,v \in {{\mathrm{{\Lambda }}}}_n\), there exists positive constant C and \({{\mathrm{{\lambda }}}}\) such that

where denotes the event that u and v are connected by a path of open edges in \(Y_t\).

Proof

Let \(\{Z_t\}\) be a second instance of the continuous time Glauber dynamics. The evolution of \(\{Z_t\}\) is coupled with that of \(\{Y_t\}\) via the identity coupling, except that \(\{Z_t\}\) never updates the edge e. The initial configuration of \(\{Z_t\}\) is sampled according to the distribution \(\mu ^{\eta }(\,\cdot \,|\,e=1)\) such that \(Y_0 \subseteq Z_0\). This is always possible because \(\mu ^{\eta }(\,\cdot \,|\,e=1) \succeq \mu ^{\eta }\). Then, \(Y_t \subseteq Z_t\) and \(Z_t\) has law \(\mu ^{\eta }(\,\cdot \,|\,e=1)\) for all \(t \ge 0\). We establish that the measure \(\mu ^{\eta }(\,\cdot \,|\,e=1)\) has exponential decay of finite volume connectivities and thus so does the distribution of \(Y_t\) for all \(t \ge 0\). By (3), for all \(u,v \in {{\mathrm{{\Lambda }}}}_n\), we have

$$\begin{aligned} \mu ^{\eta }(\,u \;{\leftrightarrow }\; v\,|\,e=1)\mu ^\eta (e=1) \le \mu ^\eta (u\;{\leftrightarrow }\; v) \le C \mathrm{e}^{-{{\mathrm{{\lambda }}}}d(u,v)}, \end{aligned}$$

where \(C,{{\mathrm{{\lambda }}}}\) are positive constants. If \(p' = \frac{p}{q(1-p)+p}\), then \(\mu ^{\eta } \succeq \mu ^{\eta }_{{{\mathrm{{\Lambda }}}},p',1}\) (see, e.g., [11]), and thus \(\mu ^\eta (e=1) \ge p'\). Since \(q \ge 1\),

$$\begin{aligned} \mu ^{\eta }(\,u\;{\leftrightarrow }\; v\,|\,e=1) \le \frac{qC}{p} \mathrm{e}^{-{{\mathrm{{\lambda }}}}d(u,v)}. \end{aligned}$$

The result then follows immediately when \(p = \varOmega (1)\). Otherwise, the measure \(\mu ^{\eta }\) is stochastically dominated by any random-cluster measure \(\mu ^{\eta }_{{{\mathrm{{\Lambda }}}},p'',q}\) with \(p''= \varOmega (1)\), for which we just established exponential decay of finite volume connectivities; the result follows by monotonicity. \(\square \)

We are now ready to prove the lemma.

Proof

Let \(Q_t\) be the random time at which the t-th edge is updated by the identity coupling. For some fixed \(\ell \) to be chosen later, and each \(t \ge 0\), consider the event

$$\begin{aligned} \mathcal{E}_{\ell ,t} := \{u \overset{\scriptscriptstyle {Y_{Q_t}}}{\nleftrightarrow } v~~~\forall u,v \in B~\text {s.t.}~d(u,v)>\ell \}, \end{aligned}$$

where \(u\overset{\scriptscriptstyle {Y_{Q_t}}}{\nleftrightarrow }v\) denotes the event that u and v are not connected by a path in \(Y_{Q_t}(B)\). Also, let \(\mathcal{E}_\ell := \bigcap _{t:Q_t \le T} \mathcal{E}_{\ell ,t}\). Then,

$$\begin{aligned} \Pr [\,X_{T}(e) \ne Y_{T}(e)\,] \le \Pr [\,X_{T}(e) \ne Y_{T}(e) \,|\, \mathcal{E}_\ell \,] + \Pr [\,\lnot \mathcal{E}_\ell \,] \end{aligned}$$
(20)

[cf. Eq. (5)]. We bound each term on the right hand side of (20) separately.

Conditioned on the event \(\mathcal{E}_\ell \), a witness for the fact that \(X_{T}(e) \ne Y_{T}(e)\) can be constructed as in discrete time. However, the probability that a given witness of length L is updated by the continuous time dynamics is instead bounded using the following fact from [16]. \(\square \)

Fact 7.3

Consider L independent rate 1 Poisson clocks. Then, the probability that there is an increasing sequence of times \(0 \le t_1< \cdots < t_L \le T\) such that clock i rings at time \(t_i\) is at most \(\left( \frac{eT}{L}\right) ^L\).

Recall from Sect. 3 that the number of witnesses of length L is at most \((4(\ell +1)^2)^L\) (crudely). Hence, following the same steps as in the proof Lemma 3.1, and taking \(\ell = r^{1/4}-1\), we get

$$\begin{aligned} \Pr [\,X_{T}(e) \ne Y_{T}(e) \,|\, \mathcal{E}_\ell \,] \le \frac{{{\mathrm{\mathrm{e}}}}}{{{\mathrm{\mathrm{e}}}}-1} \cdot {{{\mathrm{\mathrm{e}}}}}^{-r^{3/4}}, \end{aligned}$$
(21)

using the fact that \(T \le r^{1/4}/(4{{{\mathrm{\mathrm{e}}}}}^2)\) [cf. Eq. (7)].

To bound the second term on the right hand side of (20), let N be the number of edge updates in B up to time T. Observe that N is a Poisson random variable with rate \(M := |E(B,r)| = \varTheta (r^2)\). Using standard bounds for Poisson tail probabilities we get that \(\Pr [N > {{{\mathrm{\mathrm{e}}}}}^2 M T] = \exp (-\varOmega (M T))\) for all \(T \ge 1\). Therefore,

$$\begin{aligned} \Pr [\,\lnot \mathcal{E}_\ell \,] \le \Pr [\,\lnot \mathcal{E}_\ell \,|\,N \le {{{\mathrm{\mathrm{e}}}}}^2 M T\,]+{{{\mathrm{\mathrm{e}}}}}^{-\varOmega (M T)}. \end{aligned}$$

Also, \(\lnot \mathcal{E}_\ell := \bigcup _{t:Q_t \le T} \,\lnot \mathcal{E}_{\ell ,t}\), and if the edge update at time \(Q_t\) occurs outside B, we have \(\lnot \mathcal{E}_{\ell ,t} = \lnot \mathcal{E}_{\ell ,t+1}\). Hence, a union bound implies

$$\begin{aligned} \Pr [\,\lnot \mathcal{E}_\ell \,] ~\le ~ {{{\mathrm{\mathrm{e}}}}}^2 M T \max _{t:Q_t \le T} \Pr [\,\lnot \mathcal{E}_{\ell ,t}\,]+{{{\mathrm{\mathrm{e}}}}}^{-\varOmega (M T)}. \end{aligned}$$

Fact 7.2 establishes exponential decay of finite volume connectivities (3) for the distribution of \(Y_t\) in B for all \(t \ge 0\). Then, as in Lemma 3.1, we obtain

$$\begin{aligned} \Pr [\,\lnot \mathcal{E}_\ell \,] \le O(r^{6.25}) \cdot {{{\mathrm{\mathrm{e}}}}}^{-\varOmega (r^{1/4})} + {{{\mathrm{\mathrm{e}}}}}^{-\varOmega (r^2)}. \end{aligned}$$

Together with (21), this implies there exist constants \(c,C,{{\mathrm{{\lambda }}}}> 0\) such that for all \(r \ge c\) we have \(\Pr [\,X_{T}(e) \ne Y_{T}(e) \,] \le C \exp (-\lambda r^{1/4})\), as desired.

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Blanca, A., Sinclair, A. Random-cluster dynamics in \({{\mathrm{\mathbb {Z}}}}^2\) . Probab. Theory Relat. Fields 168, 821–847 (2017). https://doi.org/10.1007/s00440-016-0725-1

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Keywords

  • Random-cluster model
  • Glauber dynamics
  • Markov chains
  • Spatial mixing
  • Statistical physics

Mathematics Subject Classification

  • 60J10
  • 60K35
  • 82B20