arXiv subjects

Dmitry Ioffe

Publications and source records attributed to Dmitry Ioffe.

At least 19 recordsLinked to original sources

arXiv2024

Wetting Transition on Trees I: Percolation With Clustering

A new ``Percolation with Clustering'' (PWC) model is introduced, where (the probabilities of) site percolation configurations on the leaf set of a binary tree are rewarded exponentially according to a generic function, which measures the degree of clustering in the configuration. Conditions on such ``clustering function'' are given for the existence of a limiting free energy and a wetting transition, namely the existence of a non-trivial percolation parameter threshold above and only above which the set of ``dry'' (open) sites have an asymptotic density. Several examples of clustering functions are given and studied using the general theory. The results here will be used in a sequel paper to study the wetting transition for the discrete Gaussian free field on the tree subject to a hard wall constraint.

math.PR↗

arXiv2021

Critical prewetting in the 2d Ising model

In this paper we develop a detailed analysis of critical prewetting in the context of the two-dimensional Ising model. Namely, we consider a two-dimensional nearest-neighbor Ising model in a $2N\times N$ rectangular box with a boundary condition inducing the coexistence of the $+$ phase in the bulk and a layer of $-$ phase along the bottom wall. The presence of an external magnetic field of intensity $h=λ/N$ (for some fixed $λ>0$) makes the layer of $-$ phase unstable. For any $β>β_{\rm c}$, we prove that, under a diffusing scaling by $N^{-2/3}$ horizontally and $N^{-1/3}$ vertically, the interface separating the layer of unstable phase from the bulk phase weakly converges to an explicit Ferrari-Spohn diffusion.

math.PR↗

arXiv2021

Non-analyticity of the correlation length in systems with exponentially decaying interactions

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $\mathbb{Z}^d$ with long-range interactions of the form $J_x = ψ(x) e^{-|x|}$, where $ψ(x) = e^{\mathsf{o}(|x|)}$ and $|\cdot|$ is an arbitrary norm. We characterize explicitly the prefactors $ψ$ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $β$, magnetic field $h$, etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $ψ$ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein--Zernike theory of correlations.

math.PR↗

arXiv2020

Invariance principle for a Potts interface along a wall

We consider nearest-neighbor two-dimensional Potts models, with boundary conditions leading to the presence of an interface along the bottom wall of the box. We show that, after a suitable diffusive scaling, the interface weakly converges to the standard Brownian excursion.

math.PR↗

arXiv2020

Split-and-Merge in Stationary Random Stirring on Lattice Torus

We show that in any dimension $d\ge1$, the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson-Dirichlet law $\mathsf{PD(1)}$, as the size of the system grows to infinity. In the case of transient dimensions, $d\ge 3$, the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.

math.PR↗

arXiv2019

Tightness and Line Ensembles for Brownian Polymers under Geometric Area Tilts

We prove tightness and limiting Brownian-Gibbs description for line ensembles of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. Statistical properties of the resulting ensemble are very different from that of non-colliding Brownian bridges without self-potentials. The model itself was introduced in order to mimic level lines of (2+1)-dimensional discrete Solid-On-Solid random interfaces above a hard wall.

math.PR↗

arXiv2018

Confinement of Brownian Polymers under Geometric Area Tilts

We consider tightness for families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of $2+1$ discrete Solid-On-Solid random interfaces above a hard wall.

math.PR↗

arXiv2018

The hydrodynamic limit for local mean-field dynamics with unbounded spins

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean field interactions. We proof convergence of the empirical measure to the solution of a McKean-Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of Gärtner, Comets and others for bounded spins or strict mean field interactions.

math.PR↗

arXiv2018

Low-temperature interfaces: Prewetting, layering, faceting and Ferrari-Spohn diffusions

In this paper, we survey and discuss various surface phenomena such as prewetting, layering and faceting for a family of two- and three-dimensional low-temperature models of statistical mechanics, notably Ising models and (2+1)-dimensional solid-on-solid (SOS) models, with a particular accent on scaling regimes which lead or, in most cases, are conjectured to lead to Ferrari-Spohn type diffusions.

math.PR↗

arXiv2017

Formation of Facets for an Effective Model of Crystal Growth

We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a 2+1 solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to "grow" through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.

math.PR↗

arXiv2016

A quantitative Burton-Keane estimate under strong FKG condition

We consider translationally-invariant percolation models on $\mathbb{Z}^d$ satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance $n$ (this corresponds to a finite size version of the celebrated Burton-Keane [Comm. Math. Phys. 121 (1989) 501-505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincaré inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight $q\ge1$.

math.PR↗

arXiv2016

Dyson Ferrari--Spohn diffusions and ordered walks under area tilts

We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of $2+1$-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari--Spohn diffusions associated with limiting Sturm--Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.

math.PR↗

arXiv2015

An invariance principle to Ferrari-Spohn diffusions

We prove an invariance principle for a class of tilted (1+1)-dimensional SOS models or, equivalently, for a class of tilted random walk bridges in Z_+. The limiting objects are stationary reversible ergodic diffusions with drifts given by the logarithmic derivatives of the ground states of associated singular Sturm-Liouville operators. In the case of a linear area tilt, we recover the Ferrari-Spohn diffusion with log-Airy drift, which was derived by Ferrari and Spohn in the context of Brownian motions conditioned to stay above circular and parabolic barriers.

math.PR↗

arXiv2014

Multidimensional Random Polymers : A Renewal Approach

In these lecture notes, which are based on the mini-course given at 2013 Prague School on Mathematical Statistical Physics, we discuss ballistic phase of quenched and annealed stretched polymers in random environment on ${\mathbb Z}^d$ with an emphasis on the natural renormalized renewal structures which appear in such models. In the ballistic regime an irreducible decomposition of typical polymers leads to an effectiverandom walk reinterpretation of the latter. In the annealed casethe Ornstein-Zernike theory based on this approach paves the way to an essentially complete control on the level of local limit results and invariance principles. In the quenched case, the renewal structure maps the model of stretched polymers into an effective model of directed polymers. As a result one is able to use techniques and ideas developed in the context of directed polymers in order to address issues like strong disorder in low dimensions and weak disorder in higher dimensions. Among the topics addressed: Thermodynamics of quenched and annealed models, multi-dimensional renewal theory (under Cramer's condition), renormalization and effective random walk structure of annealed polymers, very weak disorder in dimensions $d\geq 4$ and strong disorder in dimensions $d=1,2$.

math.PR↗

arXiv2014

Interaction versus entropic repulsion for low temperature Ising polymers

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 over the pinning interaction: in particular, the contour-boundary interaction should not modify significantly the contour partition function and the related surface tension should be unchanged. Here we prove that this is indeed the case, at least at sufficiently low temperature, in a quite general framework that applies in particular to the models of interest mentioned above.

math.PR↗

arXiv2013

An Almost-Sure CLT for Stretched Polymers

We prove an almost sure CLT for spatial extension of stretched (meaning subject to a non-zero pulling force) polymers at very weak disorder in all dimensions d+1 larger than or equal to 4.

math.PR↗

arXiv2013

On the Gibbs states of the noncritical Potts model on Z^2

We prove that all Gibbs states of the q-state nearest neighbor Potts model on Z^2 below the critical temperature are convex combinations of the q pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature.

math.PR↗

arXiv2012

Ground states for mean field models with a transverse component

We investigate global logarithmic asymptotics of ground states for a family of quantum mean field models. Our approach is based on a stochastic representation and a combination of large deviation and weak KAM techniques. The spin- 1/2 case is worked out in more detail.

math.PR↗