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

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

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

"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_

A deep representation for depth images from synthetic data

Convolutional Neural Networks (CNNs) trained on large scale RGB databases have become the secret sauce in the majority of recent approaches for object categorization from RGB-D data. Thanks to colorization techniques, these methods exploit the filters learned from 2D images to extract meaningful representations in 2.5D. Still, the perceptual signature of these two kind of images is very different, with the first usually strongly characterized by textures, and the second mostly by silhouettes of objects. Ideally, one would like to have two CNNs, one for RGB and one for depth, each trained on a suitable data collection, able to capture the perceptual properties of each channel for the task at hand. This has not been possible so far, due to the lack of a suitable depth database. This paper addresses this issue, proposing to opt for synthetically generated images rather than collecting by hand a 2.5D large scale database. While being clearly a proxy for real data, synthetic images allow to trade quality for quantity, making it possible to generate a virtually infinite amount of data. We show that the filters learned from such data collection, using the very same architecture typically used on visual data, learns very different filters, resulting in depth features (a) able to better characterize the different facets of depth images, and (b) complementary with respect to those derived from CNNs pre-trained on 2D datasets. Experiments on two publicly available databases show the power of our approach

Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $\sigma$ has weight $\lambda^{|\sigma|}$, where $\lambda$ is a positive real parameter, and $|\sigma|$ is the total length of the edges in $\sigma$. Empirically, this model exhibits a "phase transition" at $\lambda=1$ (corresponding to the uniform distribution): for $\lambda<1$ distant edges behave essentially independently, while for $\lambda>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $\lambda<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $\lambda>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamics of Lattice Triangulations on Thin Rectangles

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $\lambda^{|\sigma|}$ where $\lambda>0$ is a parameter and $|\sigma|$ denotes the total edge length of the triangulation. When $\lambda\in(0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp(\Omega(n^2))$ for $\lambda>1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $\lambda=1$