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D. Ioffe

Publications and source records attributed to D. Ioffe.

9 recordsLinked to original sources
arXiv2021

Failure of Ornstein--Zernike asymptotics for the pair correlation function at high temperature and small density

We report on recent results that show that the pair correlation function of systems with exponentially decaying interactions can fail to exhibit Ornstein-Zernike asymptotics at all sufficiently high temperatures and all sufficiently small densities. This turns out to be related to a lack of analyticity of the correlation length as a function of temperature and/or density and even occurs for one-dimensional systems.

cond-mat.stat-mech↗
arXiv2013

Subcritical percolation with a line of defects

We consider the Bernoulli bond percolation process $\mathbb{P}_{p,p'}$ on the nearest-neighbor edges of $\mathbb{Z}^d$, which are open independently with probability $p p_c'$. Moreover, $p_c'(p,2)=p_c'(p,3)=p$, and $p_c'(p,d)>p$ for $d\geq 4$. We also analyze the behavior of $ξ_p-ξ_{p,p'}$ as $p'\downarrow p_c'$ in dimensions $d=2,3$. Finally, we prove that when $p'>p_c'$, the following purely exponential asymptotics holds: \[\mathbb {P}_{p,p'}(0\leftrightarrow n\mathbf {e}_1)=ψ_de^{-ξ_{p,p'}n}\bigl(1+o(1)\bigr)\] for some constant $ψ_d=ψ_d(p,p')$, uniformly for large values of $n$. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and don't rely on exact computations.

math.PR↗
arXiv2008

Sharp asymptotics for metastability in the random field Curie-Weiss model

In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.

math.PR↗
arXiv2003

Random path representation and sharp correlations asymptotics at high-temperatures

We recently introduced a robust approach to the derivation of sharp asymptotic formula for correlation functions of statistical mechanics models in the high-temperature regime. We describe its application to the nonperturbative proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the proof, in the Ising case, to arbitrary odd-odd correlation functions. We discuss the fluctuations of connection paths (invariance principle), and relate the variance of the limiting process to the geometry of the equidecay profiles. Finally, we explain the relation between these results from Statistical Mechanics and their counterparts in Quantum Field Theory.

math.PR↗
arXiv2001

Ornstein-Zernike Theory for the finite range Ising models above T_c

We derive precise Ornstein-Zernike asymptotic formula for the decay of the two-point function in the general context of finite range Ising type models on Z^d. The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernandez, goes through in the whole of the high temperature region T > T_c. As a byproduct we obtain that for every T > T_c, the inverse correlation length is an analytic and strictly convex function of direction.

math.PR↗
arXiv2001

Winterbottom Construction for Finite Range Ferromagnetic Models: An L_1 Approach

We provide a rigorous microscopic derivation of the thermodynamic description of equilibrium crystal shapes in the presence of a substrate, first studied by Winterbottom. We consider finite range ferromagnetic Ising models with pair interactions in dimensions greater or equal to 3, and model the substrate by a finite-range boundary magnetic field acting on the spins close to the bottom wall of the box.

math.PR↗
arXiv1999

Rigorous Probabilistic Analysis of Equilibrium Crystal Shapes

The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

math.PR↗
arXiv1998

A Note on the Decay of Correlations Under $δ$-Pinning

We prove that for a class of massless $\nablaϕ$ interface models on $\Ztwo$ an introduction of an arbitrary small pinning self-potential leads to exponential decay of correlation, or, in other words, to creation of mass.

math.PR↗