We study zero-temperature Glauber dynamics on \Z^d, which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define p_c(\Z^d) to be the infimum over p such that the system fixates at '+' with probability 1. It is a folklore conjecture that p_c(\Z^d) = 1/2 for every 2 \le d \in \N. We prove that p_c(\Z^d) \to 1/2 as d \to \infty.
In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature $β$ ranges from 0 to $β_0=\frac12\mathrm{arctanh}(\frac13)$. Moreover…
We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with plus-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield …
The wavefunction of a single spin system in a prepared initial state evolves to equilibrium with a heat bath. The average spin $$q(t) = p_{\uparrow}(t) - p_{\downarrow}(t)$$ exhibits a characteristic time for this evolution. With the proper choice of spin flip rates, a dynamical Ising model (Glauber) can be constructed with the same characteristic time for transition of the average spin to equilibrium. The Glauber dynamics are expressed as a Markoff process that possesses many of the same physical characteristics as its quantum mechanical counterpart. In addition, since the classical trajector…
We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative e…
In this paper, we prove a general result concerning finite-range, attractive interacting particle systems on $\{-1, 1\}^{\mathbb{Z}^d}$. If the particle system has a unique stationary measure and, in a precise sense, relaxes to this stationary measure at an exponential rate then any small perturbation of the dynamics also has a unique stationary measure to which it relaxes at an exponential rate. To augment this result, we study the particular case of Glauber dynamics for the Ising model. We show that for any non-zero external field the dynamics converges to its unique invariant measure at an …
It has been recently shown by H.Weber and J.C. Mourrat, for the two-dimensional Ising-Kac model at critical temperature, that the fluctuation field of the magnetization, under the Glauber dynamic, converges in distribution to the solution of a non linear ill-posed SPDE: the dynamical $Φ^4_2$ equation. In this article we consider the case of the multivatiate stochastic quantization equation $Φ^{2n}_2$ on the two-dimensional torus, and we answer to a conjecture of H.Weber and H.Shen. We show that it is possible to find a state space for a spin system on the two-dimensional discrete torus undergo…
We study a nonequilibrium mean field Ising model in the low temperature phase regime, where metastable equilibrium states develop a cuspidal (spinodal) singularity. We focus on celebrated Glauber dynamics, and design a contact Hamiltonian flow which captures some of its rough features in this regime. We prove, however, that there is an inevitable discrepancy between the scaling laws for the relaxation time in the Glauber and the contact Hamiltonian dynamical systems.
We study Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss Model. It is well known that at high temperature ($β< 1$) the mixing time is $Θ(n\log n)$, whereas at low temperature ($β> 1$) it is $\exp(Θ(n))$. Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed $β> 1$, the mixing-time of this model is $Θ(n\log n)$, analogous to the high-temperature regime of the original dynamics. Furthermore, they showed \emph{cutoff} for the original dynam…
A fast harmonic oscillator is linearly coupled with a system of Ising spins that are in contact with a thermal bath, and evolve under a slow Glauber dynamics at dimensionless temperature $θ$. The spins have a coupling constant proportional to the oscillator position. The oscillator-spin interaction produces a second order phase transition at $θ=1$ with the oscillator position as its order parameter: the equilibrium position is zero for $θ>1$ and non-zero for $θ< 1$. For $θ<1$, the dynamics of this system is quite different from relaxation to equilibrium. For most initial conditions, the oscill…
We study the Glauber dynamics of a two dimensional Blume-Capel model (or dilute Ising model) with Kac potential parametrized by $(β,θ)$ - the "inverse temperature" and the "chemical potential". We prove that the locally averaged spin field rescales to the solution of the dynamical $Φ^4$ equation near a curve in the $(β,θ)$ plane and to the solution of the dynamical $Φ^6$ equation near one point on this curve. Our proof relies on a discrete implementation of Da Prato-Debussche method as in a result by Mourrat-Weber but an additional coupling argument is needed to show convergence of the lineari…
We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree ($Δ$) of the underlying graph and the number of colours or spins ($q$) in determining whether the dynamics mixes rapidly or not. We find a lower bound $L$ on the number of colours such that Glauber dynamics is rapidly mixing if at least $L$ colours are used. We give a closely-matching upper bound $U$ on the number of colours such that with probability th…
We consider an equilibrium birth and death type process for a particle system in infinite volume, the latter is described by the space of all locally finite point configurations on $\R^d$. These Glauber type dynamics are Markov processes constructed for pre-given reversible measures. A representation for the "carré du champ" and "second carré du champ" for the associate infinitesimal generators $L$ are calculated in infinite volume and a corresponding coercivity identity is derived. The latter is used to give explicit sufficient conditions for the appearance and bounds for the size of the spec…
We consider proper 5-colourings of the kagome lattice. Proper q-colourings correspond to configurations in the zero-temperature q-state anti-ferromagnetic Potts model. Salas and Sokal have given a computer assisted proof of strong spatial mixing on the kagome lattice for q>=6 under any temperature, including zero temperature. It is believed that there is strong spatial mixing for q>=4. Here we give a computer assisted proof of strong spatial mixing for q=5 and zero temperature. It is commonly known that strong spatial mixing implies that there is a unique infinite-volume Gibbs measure and that…
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb{R}^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $mu$ as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of ``infinite length'' will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in $\mathbb{R}^d$). We prove that, in the low activity-high temperature …
We study the stochastic Ising model on finite graphs with n vertices and bounded degree and analyze the effect of boundary conditions on the mixing time. We show that for all low enough temperatures, the spectral gap of the dynamics with (+)-boundary condition on a class of nonamenable graphs, is strictly positive uniformly in n. This implies that the mixing time grows at most linearly in n. The class of graphs we consider includes hyperbolic graphs with sufficiently high degree, where the best upper bound on the mixing time of the free boundary dynamics is polynomial in n, with exponent growi…
We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with $q\geq 3$ states and show that it undergoes a critical slowdown at an inverse-temperature $β_s(q)$ strictly lower than the critical $β_c(q)$ for uniqueness of the thermodynamic limit. The dynamical critical $β_s(q)$ is the spinodal point marking the onset of metastability. We prove that when $β<β_s(q)$ the mixing time is asymptotically $C(β, q) n \log n$ and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order $n$. At $β=β_s(q)$ the dynamics no longer exhibits cutoff and its …
We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the $4$-dimensional $n$-component $|φ|^4$ model at the critical point and its approach from the high temperature side, and of the $2$-dimensional Sine-Gordon and the Discrete Gaussian m…
We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than $1$. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds fo…
The six-vertex model in statistical physics is a weighted generalization of the ice model on $\mathbb{Z}^2$ (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require…
A harmonic oscillator linearly coupled with a linear chain of Ising spins is investigated. The $N$ spins in the chain interact with their nearest neighbours with a coupling constant proportional to the oscillator position and to $N^{-1/2}$, are in contact with a thermal bath at temperature $T$, and evolve under Glauber dynamics. The oscillator position is a stochastic process due to the oscillator-spin interaction which produces drastic changes in the equilibrium behaviour and the dynamics of the oscillator. Firstly, there is a second order phase transition at a critical temperature $T_c$ whos…
We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability $p \in (0,1)$ and a cluster weight $q > 0$. We establish that for every $q\ge 1$, the random-cluster Glauber dynamics mixes in optimal $Θ(n\log n)$ steps on $n$-vertex random graphs having a prescribed degree sequence with bounded average branching $γ$ throughout the full high-temperature uniqueness regime $p<p_u(q,γ)$. The family of random graph mode…
We investigate the effect of disorder on the Curie-Weiss model with Glauber dynamics. In particular, we study metastability for spin-flip dynamics on the Erdős-Rényi random graph $ER_n(p)$ with $n$ vertices and with edge retention probability $p \in (0,1)$. Each vertex carries an Ising spin that can take the values $-1$ or $+1$. Single spins interact with an external magnetic field $h \in (0,\infty)$, while pairs of spins at vertices connected by an edge interact with each other with ferromagnetic interaction strength $1/n$. Spins flip according to a Metropolis dynamics at inverse temperature …
We consider the 2D stochastic Ising model evolving according to the Glauber dynamics at zero temperature. We compute the initial drift for droplets which are suitable approximations of smooth domains. A specific spatial average of the derivative at time~0 of the volume variation of a droplet close to a boundary point is equal to its curvature multiplied by a direction dependent coefficient. We compute the explicit value of this coefficient.
We consider several models whose motivation arises from statistical mechanics. We begin by investigating some families of distributions of translation invariant subgraphs of some Cayley graphs, and in particular subgraphs of the square lattice. We then discuss some properties of the Spin-Glass model in that lattice. We continue in describing some properties of the Spin-Glass models in some other graphs. The last two parts of this work are devoted to the understanding of two dynamical processes on graphs. The first one is the well known zero-temperature Glauber dynamics on some families of grap…
We study dynamical aspects of the $q$-state Potts model on an $n\times n$ box at its critical $β_c(q)$. Heat-bath Glauber dynamics and cluster dynamics such as Swendsen--Wang (that circumvent low-temperature bottlenecks) are all expected to undergo "critical slowdowns" in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is $O(1)$, should at criticality be polynomial in $n$ for $1< q \leq 4$, and exponential in $n$ for $q>4$ in accordance with the predicted discontinuous phase transition. This was confirmed for $q=2$ (the Ising model) by th…
We study the Glauber dynamics for the zero-temperature Ising model in dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an hypercube of size L entirely filled with "minus" spins to become entirely "plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called "Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key point of our proof is to use the detail knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces …
A new scheme is proposed to design excited coherent states. where the states $β$,$α$ denote the Glauber two variable minimum uncertainty coherent states, which minimize minimum uncertainty conditions while carrier nonclassical properties too and n is an integer. They are converted into the Agarwal's type of the photon added coherent states, arbitrary Fock states and the Glauber two variable coherent states, respectively depending on which of the parameters $\b{eta}$,$α$ and $n$ equal to zero. It has been shown that the resolution of identity condition is realized with respect to an appropriate…
In this study, we investigate the energy landscape of the Ising and Potts models on fixed and finite but large three-dimensional (3D) lattices where no external field exists and quantitatively characterize the metastable behavior of the associated Glauber dynamics in the very low temperature regime. Such analyses for the models with non-zero external magnetic fields have been extensively performed over the past two decades; however, models without external fields remained uninvestigated. Recently, the corresponding investigation has been conducted for the two-dimensional (2D) model without an …
Dynamical hysteresis is a phenomenon which arises in ferromagnetic systems below the critical temperature as a response to adiabatic variations of the external magnetic field. We study the problem in the context of the mean-field Ising model with Glauber dynamics, proving that for frequencies of the magnetic field oscillations of order $N^{2/3}$, with $N$ the size of the system, the "critical" hysteresis loop becomes random.
