Relaxation time of anisotropic simple exclusion processes and quantum Heisenberg models

Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with $N$ particles in a rectangle of \bbZ^2. Every particle at row $h$ tries to jump to an arbitrary empty site at row $h\pm 1$ with rate $q^{\pm 1}$, where $q\in (0,1)$ is a measure of the drift driving the particles towards the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in $N$ and in the size of the rectangle. The proof is inspired by a recent interesting technique envisaged by E. Carlen, M.C. Carvalho and M. Loss to analyze the Kac model for the non linear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin--S, {\rm S}\in \frac12\bbN, XXZ chain and for the 111 interface of the spin--S XXZ Heisenberg model, thus generalizing previous results valid only for spin $\frac12$.Comment: 27 page

On the approach to equilibrium for a polymer with adsorption and repulsion

We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path \eta has a weight \lambda^{N(\eta)}, where \lambda>0 and N(\eta) is the number of zeros in \eta. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength \lambda is varied. In this paper we study a natural 'spin flip' dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (\lambda=1, no wall), where the gap and the mixing time are known to scale as L^{-2} and L^2\log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for \lambda \geq 1 the relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (\lambda < 1) the gap is shown to be O(L^{-5/2}), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.Comment: 43 pages, 5 figures; v2: corrected typos, added Table

On the probability of staying above a wall for the (2+1)-dimensional SOS model at low temperature

We obtain sharp asymptotics for the probability that the (2+1)-dimensional discrete SOS interface at low temperature is positive in a large region. For a square region $\Lambda$, both under the infinite volume measure and under the measure with zero boundary conditions around $\Lambda$, this probability turns out to behave like $\exp(-\tau_\beta(0) L \log L )$, with $\tau_\beta(0)$ the surface tension at zero tilt, also called step free energy, and $L$ the box side. This behavior is qualitatively different from the one found for continuous height massless gradient interface models.Comment: 21 pages, 6 figure

Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let $L$ be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the $\hat z$ axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as $L\to \infty$. In \cite{BCNS} it was proved that, by perturbing in a sub--cylinder with basis of linear size $R\ll L$ the interface ground state, it is possible to construct excited states whose energy gap shrinks as $R^{-2}$. Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from below by $\text{const.}L^{-2}$. We prove the result by first mapping the problem into an asymmetric simple exclusion process on $\Z^3$ and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in \cite{CancMart}. Along the way we improve some bounds on the equivalence of ensembles already discussed in \cite{BCNS} and we establish an upper bound on the density of states close to the bottom of the spectrum.Comment: 48 pages, latex2e fil

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup $P_t$. A fundamental and still largely open problem is the understanding of the long time behavior of \d_\h P_t when the initial configuration \h is sampled from a highly disordered state $\nu$ (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular $b$-ary tree \Tree^b, we tackle the above problem for the Ising and hard core gas (independent sets) models on \Tree^b. If $\nu$ is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove $\nu$-almost sure weak convergence of \d_\h P_t to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time $t$. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.Comment: 35 page

"Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion

We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an initial domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].Comment: 44 pages, 7 figures. v2: Theorem 1 improved to include a matching lower bound on tau_

Time scale separation in the low temperature East model: Rigorous results

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbour is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. Specifically we analyse time scale separation and dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-\gamma})$, $\gamma<1$, we show that the characteristic time scale associated to two length scales $d/q^\gamma$ and $d'/q^\gamma$ are indeed separated by a factor $q^{-a}$, $a=a(\gamma)>0$, provided that $d'/d$ is large enough independently of $q$. In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. Finally we show that no form of time scale separation can occur for $\gamma=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was previously assumed in the physical literature based on numerical simulations.Comment: 6 pages, 0 figures; clarified q dependence of bounds, results unchange

Harmonic pinnacles in the Discrete Gaussian model

The 2D Discrete Gaussian model gives each height function $\eta : \mathbb{Z}^2\to\mathbb{Z}$ a probability proportional to $\exp(-\beta \mathcal{H}(\eta))$, where $\beta$ is the inverse-temperature and $\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2$ sums over nearest-neighbor bonds. We consider the model at large fixed $\beta$, where it is flat unlike its continuous analog (the Gaussian Free Field). We first establish that the maximum height in an $L\times L$ box with 0 boundary conditions concentrates on two integers $M,M+1$ with $M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}$. The key is a large deviation estimate for the height at the origin in $\mathbb{Z}^2$, dominated by "harmonic pinnacles", integer approximations of a harmonic variational problem. Second, in this model conditioned on $\eta\geq 0$ (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels $H,H+1$ where $H\sim M/\sqrt{2}$. This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and Fr\"ohlich (1986), where it was argued that the maximum and the height of the surface above a floor are both of order $\sqrt{\log L}$. Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to $p$-harmonic analysis and alternating sign matrices.Comment: 40 pages, 5 figure