Separated-occurrence inequalities are variants for dependent lattice models of the van den Berg-Kesten inequality for independent models. They take the form $P(A \circ_r B) \leq (1 + ce^{-εr})P(A)P(B)$, where $A \circ_r B$ is the event that $A$ and $B$ occur at separation $r$ in a configuration $ω$, that is, there exist two random sets of bonds or sites separated by at least distance $r$, one set responsible for the occurrence of the event $A$ in $ω$, the other for the occurrence of $B$. We establish such inequalities for certain subcritical FK models, and for certain Ising models which are at supercritical temperature or have an external field, with $A$ and $B$ increasing or decreasing events.
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)
We reconsider the percolation approach of Russo, Aizenman and Higuchi for showing that there exist only two phases in the Ising model on the square lattice. We give a fairly short alternative proof which is only based on FKG monotonicity and avoids the use of GKS-type inequalities originally needed for some background results. Our proof extends to the Ising model on other planar lattices such as the triangular and honeycomb lattice. We can also treat the Ising antiferromagnet in an external field and the hard-core lattice gas model on $Z^2$.
We study constrained percolation models on planar lattices including the $[m,4,n,4]$ lattice and the square tilings of the hyperbolic plane, satisfying certain local constraints on faces of degree 4, and investigate the existence of infinite clusters. The constrained percolation models on these lattices are closely related to Ising models and XOR Ising models on regular tilings of the Euclidean plane or the hyperbolic plane. In particular, we obtain a complete picture of the number of infinite "$+$" and "$-$" clusters of the ferromagnetic Ising model with the free boundary condition on a verte…
The Ising model at inverse temperature $β$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-β}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $β$, this generates a dependent (spin) percolation model with parameter $r$. We show…
We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $β<β_c$, and the mean-field lower bound $\mathbb{P}_β[0\longleftrightarrow\infty]\ge (β-β_c)/β$ for $β>β_c$. For finite-range models, we also prove that for any $β<β_c$, the probability of an open path from the origin to distance $n$ decays exponentially fast in $n$. For the Ising model, we pro…
We introduce a new framework for analyzing Glauber dynamics for the Ising model. The traditional approach for obtaining sharp mixing results has been to appeal to estimates on spatial properties of the stationary measure from within a multi-scale analysis of the dynamics. Here we propose to study these simultaneously by examining "information percolation" clusters in the space-time slab. Using this framework, we obtain new results for the Ising model on $(\mathbb{Z}/n\mathbb{Z})^d$ throughout the high temperature regime: total-variation mixing exhibits cutoff with an $O(1)$-window around the t…
For the FK representation of the Ising model, we prove that the slab percolation threshold coincides with the critical temperature in any dimension larger or equal to three.
We study the 2d-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence.
We consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at/near critical external fields. We show that all scaling relations, except a single hyperscaling relation, hold under the power law assumptions for the one-arm path and four-arm paths.
We continue our study of exponential law for occurrences and returns of patterns in the context of Gibbsian random fields. For the low temperature plus phase of the Ising model, we prove exponential laws with error bounds for occurrence, return, waiting and matching times. Moreover we obtain a Poisson law for the number of occurrences of large cylindrical events and a Gumbel law for the maximal overlap between two independent copies. As a by-product, we derive precise fluctuation results for the logarithm of waiting and return times. The main technical tool we use, in order to control mixing, …
We consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at or near critical external fields. The incipient infinite cluster (IIC) measure in the sense of Kesten is constructed. As a consequence, we can obtain some geometric properties of IIC. The result holds also for the triangular lattice.
We consider the standard first passage percolation model on $\mathbb Z^ d$ with a distribution $G$ taking two values $0<a<b$. We study the maximal flow through the cylinder $[0,n]^ {d-1}\times [0,hn]$ between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in $O(\frac {n^{d-1}} {\log n})$, for $h\geq h_0$ (for a large enough constant $h_0=h_0(a,b)$). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder $[0,n]^ {d-1}\times [0,hn]$ is superconcentrated when o…
We study infinite ``$+$'' or ``$-$'' clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph $G$ with finite vertex degree. If the critical percolation probability $p_c^{site}$ for the i.i.d.~Bernoulli site percolation on $G$ is less than $\frac{1}{2}$, we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite ``$+$''-clusters and infinitely many infinite ``$-$''-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If $p_c^{site}>\frac{1}{2}$, we obtain …
Information percolation is a new method for analyzing stochastic spin systems through classifying and controlling the clusters of information-flow in the space-time slab. It yielded sharp mixing estimates (cutoff with an $O(1)$-window) for the Ising model on $Z^d$ up to the critical temperature, as well as results on the effect of initial conditions on mixing. In this expository note we demonstrate the method on lattices (more generally, on any locally-finite transitive graph) at very high temperatures.
Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising mode…
We prove, using the random-cluster model, a strict inequality between site percolation and magnetization in the region of phase transition for the d-dimensional Ising model, thus improving a result of [CNPR76]. We extend this result also at the case of two plane lattices Z^2x{0,1} (slabs) and give a characterization of phase transition in this case. The general case of N slabs, with N an arbitrary positive integer, is partially solved and it is used to show that this characterization holds in the case of three slabs with periodic boundary conditions. However in this case we do not obtain usefu…
We provide non-asymptotic $L^1$ bounds to the normal for four well-known models in statistical physics and particle systems in $\mathbb{Z}^d$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^1$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for …
One of the most well-known classical results for site percolation on the square lattice is the equation p_c + p_c^* = 1. In words, this equation means that for all values different from p_c of the parameter p the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed `star' cluster. This result is closely related to the percolation transition being sharp: Below p_c the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distrubtion with an exponential tail. The analog of this result has been proved by Higuchi in 1993 for t…
One of the most well-known classical results for site percolation on the square lattice is the equation $p_c+p_c^*=1$. In words, this equation means that for all values $\neq p_c$ of the parameter $p$, the following holds: either a.s. there is an infinite open cluster or a.s. there is an infinite closed "star" cluster. This result is closely related to the percolation transition being sharp: below $p_c$, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in 1993 for two-…
In this paper, we investigate the behaviour of statistical physics models on a book with pages that are isomorphic to half-planes. We show that even for models undergoing a continuous phase transition on $\mathbb Z^2$, the phase transition becomes discontinuous as soon as the number of pages is sufficiently large. In particular, we prove that the Ising model on a three pages book has a discontinuous phase transition (if one allows oneself to consider large coupling constants along the line on which pages are glued). Our work confirms predictions in theoretical physics which relied on renormali…
We investigate the problem of constructing a dynamics on edge--spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations. We dream of constructing a Markov process on edge--spin configurations which is reversible with respect to the Ising--FK coupling measure, and such that the marginal on the spins is a Glauber dynamics, while the marginal on the edges is a Markovian evolution. We present two local dynamics, one which fulfills only the first condition and one which fulfills the first two conditions. …
We prove that for the Ising model defined on the plane $\Z^2$ at $β=β_c$, the average magnetization under an external magnetic field $h>0$ behaves exactly like \[{σ_0}_{β_c, h} \asymp h^{\frac 1 {15}}\,. \] The proof, which is surprisingly simple compared to an analogous result for percolation (i.e. that $θ(p)=(p-p_c)^{5/36+o(1)}$ on the triangular lattice \cite{\SmirnovWerner,\KestenScaling}) relies on the GHS inequality as well as the RSW theorem for FK percolation from \cite{\RSWfk}. The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it …
We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also…
The contact model for the spread of disease may be viewed as a directed percolation model on $\ZZ \times \RR$ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on $\ZZ \times \RR$, with associated continuum Ising and Potts models. These models are of i…
Stochastic dominations for FK percolation and sharp thinning thresholds for the Ising energy field
Université Claude Bernard Lyon 1（里昂第一大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心联合研究实验室5208，卡米尔·若当研究所）；Universidad de Chile（智利大学）；Centro de Modelamiento Matemático (AFB170001)（数学建模中心）；UMI-CNRS 2807（中法科研联合单位2807）
We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model. We also point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian Free Field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model)…
Consider a Markov process ω_t at equilibrium and some event C (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation \Pr[ω_0,ω_t \in C] - \Pr[ω_0 \in C]^2. A second natural measure is the probability of the continual occurrence event \{ω_s \in C, \forall s\in [0,t]\}. We show that for reversible Markov chains, and any event C, pairwise decorrelation of the event C implies a decay of the probability of the continual occurrence event \{ω_s \in C, \forall s \in [0,t]\} as t\to\infty. We provide examples showing that our results a…
Let $G$ be the product of finitely many trees $T_1\times T_2 \times \cdots \times T_N$, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new. We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function alon…
We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLE$_κ( κ-6)$ with $κ=16/3$. Our proof follows the classical excursion-construction of SLE$_κ(κ-6)$ processes in the continuum and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from [KS15, KS16] as it only relies on the convergence to the chordal SLE$_κ$ process in Dobrushin…
In this paper we will investigate dynamic stability of percolation for the stochastic Ising model and the contact process. We also introduce the notion of downward and upward $ε$-movability which will be a key tool for our analysis.
We prove Cardy's formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zero-temperature case of Domany's stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices); it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors.
Phase separation is a fairly common physical phenomenon with examples including the formation of water droplets from humid air (fog, rain), the separation of a crystalline structure from an isotropic material such as a liquid or even the formation of the sizzling gas bubbles when a soda can is opened. It was recognized long ago (at least on a phenomenological level) that systems exhibiting several phases in equilibrium can be described with an appropriate variational principle: the phases arrange themselves in such a way that the energy associated with the phase boundaries is minimal. Typicall…
In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension d > 1, then p_c(G) < 1. This gives a partial answer to a question of Benjamini and Schramm. As a consequence of this result we derive, under the additional condition of bounded degree, that these graphs also undergo a non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and the beach model. Our techniques are also applied to dependent percolation…
