Consider the centered Gaussian field on the lattice $\mathbb{Z}^d,$ $d$ large enough, with covariances given by the inverse of $\sum_{j=k}^K q_j(-Δ)^j,$ where $Δ$ is the discrete Laplacian and $q_j \in \mathbb{R},k\leq j\leq K,$ the $q_j$ satisfying certain additional conditions. We extend a previously known result to show that the probability that all spins are nonnegative on a box of side-length $N$ has an exponential decay at rate of order $N^{d-2k}\log{N}.$ The constant is given in terms of a higher-order capacity of the unit cube, analogous to the known case of the lattice free field. This result then allows us to show that, if we condition the field to stay positive in the $N-$box, the local sample mean of the field is pushed to a height of order $\sqrt{\log N}.$
The dynamic entropic repulsion for the Ginzburg-Landau $\nablaϕ$ interface model was discussed in [Deuschel-N. 2007] and the asymptotics of the height of the interface was identified. This paper studies a similar problem for two interfaces on the wall which are interacting with one another by the exclusion rule. Each leading order of the asymptotics of height is $\sqrt{\log t}$ as $t\to\infty$ for the system on $\integer^d,\,d\ge3$, $\log t$ for the system on $\integer^2$. The coefficient of the leading term for each interface is also identified.
We investigate the extent to which the phenomenon of Brownian entropic repulsion is universal. Consider a Brownian motion conditioned on the event $\mathcal{E}$ -- that its local time is bounded everywhere by 1. This event has probability zero and so must be approximated by events of positive probability. We prove that several natural quantities, in particular the speed of the process, are highly sensitive to the approximation procedure, and hence are not universal. However, we also propose an exponent $κ$ -- which measures the strength of the entropic repulsion by evaluating the probability t…
Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin
Bar-Ilan University（巴伊兰大学）；Einstein Institute of Mathematics, Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）；School of Mathematics, Monash University（蒙纳士大学数学学院）
We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nablaϕ|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $Λ$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $Λ$, the probability that the surface is nonnegative in $Λ$ behaves like $\exp(-4βτ_{p,β} L H_p(L) )$, where $β$ is the inverse temperature, $τ_{p,β}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic…
We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.
We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.
Contours associated to many interesting low-temperature statistical mechanics models (2D Ising model, (2+1)D SOS interface model, etc) can be described as self-interacting and self-avoiding walks on $\mathbb Z^2$. When the model is defined in a finite box, the presence of the boundary induces an interaction, that can turn out to be attractive, between the contour and the boundary of the box. On the other hand, the contour cannot cross the boundary, so it feels entropic repulsion from it. In various situations of interest a crucial technical problem is to prove that entropic repulsion prevails …
We consider an interface above an attractive hard wall in the complete wetting regime, and submitted to the action of an external increasing, convex potential, and study its delocalization as the intensity of this potential vanishes. Our main motivation is the analysis of critical prewetting, which corresponds to the choice of a linear external potential. We also present partial results on critical prewetting in the two dimensional Ising model, as well as a few (weak) results on pathwise estimates for the pure wetting problem for effective interface models.
We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open a…
We study the Glauber dynamics for the $(2+1)\mathrm{D}$ Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L\times L$ box of $\mathbb{Z}^2$ with zero boundary conditions, at large inverse-temperature $β$. It was shown by Bricmont, El Mellouki and Fröhlich [J. Stat. Phys. 42 (1986) 743-798] that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height $H\asymp(1/β)\log L$. As an essential step in understanding the effect of entropic repulsion on the Glauber dynamics we determine the equilibrium height $H$ to within an additi…
Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\mathbb{Z}^d$, $d \geq 3$, to the …
We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $Z^d$, d bigger or equal to three, below a given level disconnects the discrete blow-up of a compact set A from the boundary of the discrete blow-up of a box that contains A, when the level set of the Gaussian free field above this level is in a strongly percolative regime. These bounds substantially strengthen the results of arXiv:1412.3960, where A was a box and the convexity of A played an important role in the proof. We also derive an asymptotic upper bound on the…
We investigate level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level $α$ disconnects the discrete blow-up of a compact set $A$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of $A$, when disconnection occurs. These bounds, combined with the findings of the recent article [12],…
We study level-set percolation for the harmonic crystal on $\mathbb{Z}^d$, $d \geq 3$, with uniformly elliptic random conductances. We prove that this model undergoes a non-trivial phase transition at a critical level that is almost surely constant under the environment measure. Moreover, we study the disconnection event that the level-set of this field below a level $α$ disconnects the discrete blow-up of a compact set $A \subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We obtain quenched asymptotic upper and lower bounds on its probability in terms of the homogenized capacity o…
We consider the harmonic crystal on the d-dimensional lattice, d larger or equal to 3, that is the centered Gaussian field $ϕ$ with covariance given by the Green function of the simple random walk on $Z^d$. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition the field to be larger than an IID field $σ$ (which is also independent of $ϕ$), for every x in a large region $D_N=ND\cap \Z^d$, with N a positive integer and $D \subset\R^d$. We are mostly motivated by results for given typical realizations of the $σ$ (quenched set-up), since the c…
We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a h…
We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative. For example, we compute the width of the largest downward spike of a given length. Through the prism of these results, we find that the extrema of the free field under entropic repulsion (minus its mean) and those of the unconditioned free field are identical. Finally, when compared to previous results these findings reveal a s…
A marginal problem asks whether a given family of marginal distributions for some set of random variables arises from some joint distribution of these variables. Here we point out that the existence of such a joint distribution imposes non-trivial conditions already on the level of Shannon entropies of the given marginals. These entropic inequalities are necessary (but not sufficient) criteria for the existence of a joint distribution. For every marginal problem, a list of such Shannon-type entropic inequalities can be calculated by Fourier-Motzkin elimination, and we offer a software interfac…
We study the local structure of the extremal process associated with the Discrete Gaussian Free Field (DGFF) in scaled-up (square-)lattice versions of bounded open planar domains subject to mild regularity conditions on the boundary. We prove that, in the scaling limit, this process tends to a Cox process decorated by independent, correlated clusters whose distribution is completely characterized. As an application, we control the scaling limit of the discrete supercritical Liouville measure, extract a Poisson-Dirichlet statistics for the limit of the Gibbs measure associated with the DGFF and…
It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirem…
