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Diana Conache

Publications and source records attributed to Diana Conache.

7 recordsLinked to original sources
arXiv2023

Doeblin measures: uniqueness and mixing properties

In this paper we solve two open problems in ergodic theory. We prove first that if a Doeblin function $g$ (a $g$-function) satisfies \[\limsup_{n\to\infty}\frac{\mbox{var}_n \log g}{n^{-1/2}} < 2,\] then we have a unique Doeblin measure ($g$-measure). This result indicates a possible phase transition in analogy with the long-range Ising model. Secondly, we provide an example of a Doeblin function with a unique Doeblin measure that is not weakly mixing, which implies that the sequence of iterates of the transfer operator does not converge, solving a well-known folklore problem in ergodic theory. Previously it was only known that uniqueness does not imply the Bernoulli property.

math.PR↗
arXiv2021

Variance of voltages in a lattice Coulomb gas

We study the behavior of the variance of the difference of energies for putting an additional electric unit charge at two different locations in the two-dimensional lattice Coulomb gas in the high-temperature regime. For this, we exploit the duality between this model and a discrete Gaussian model. Our estimates follow from a spontaneous symmetry breaking in the latter model.

math.PR↗
arXiv2019

Dislocation lines in three-dimensional solids at low temperature

We propose a model for three-dimensional solids on a mesoscopic scale with a statistical mechanical description of dislocation lines in thermal equilibrium. The model has a linearized rotational symmetry, which is broken by boundary conditions. We show that this symmetry is spontaneously broken in the thermodynamic limit at small positive temperatures.

math-ph↗
arXiv2018

One-sided continuity properties for the Schonmann projection

We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In $1989$ Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure, the question whether or not it is a Gibbs measure in an almost sure sense remains open. In this paper we study the same measure by interpreting it as a temporal process. One of our main results is that the Schonmann projection is almost surely a regular $g$-measure. That is, it does possess the corresponding one-sided notion of almost Gibbsianness. We further deduce strong one-sided mixing properties which are of independent interest. Our proofs make use of classical coupling techniques and some monotonicity properties which are known to hold for one-sided, but not two-sided conditioning for FKG measures.

math.PR↗
arXiv2017

Gibbs Measures on Marked Configuration Spaces: Existence and Uniqueness

We study equilibrium states of an infinite system of interacting particles in a Euclidean space. The particles bear `unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is pairwise and splits into position-position and spin-spin parts. The position-position part is described by a superstable potential, and the spin-spin part is attractive and of finite range. Thermodynamic states of the system are defined as tempered Gibbs measures on the space of marked configurations. We derive sufficient conditions of the existence and uniqueness of these Gibbs measures.

math-ph↗
arXiv2015

Equilibrium diffusion on the cone of discrete Radon measures

Let $\mathbb K(\mathbb R^d)$ denote the cone of discrete Radon measures on $\mathbb R^d$. There is a natural differentiation on $\mathbb K(\mathbb R^d)$: for a differentiable function $F:\mathbb K(\mathbb R^d)\to\mathbb R$, one defines its gradient $\nabla^{\mathbb K} F $ as a vector field which assigns to each $η\in \mathbb K(\mathbb R^d)$ an element of a tangent space $T_η(\mathbb K(\mathbb R^d))$ to $\mathbb K(\mathbb R^d)$ at point $η$. Let $ϕ:\mathbb R^d\times\mathbb R^d\to\mathbb R$ be a potential of pair interaction, and let $μ$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $\mathbb R^d$. In particular, $μ$ is a probability measure on $\mathbb K(\mathbb R^d)$ such that the set of atoms of a discrete measure $η\in\mathbb K(\mathbb R^d)$ is $μ$-a.s.\ dense in $\mathbb R^d$. We consider the corresponding Dirichlet form $$ \mathscr E^{\mathbb K}(F,G)=\int_{\mathbb K(\mathbb R^d)}\langle\nabla^{\mathbb K} F(η), \nabla^{\mathbb K} G(η)\rangle_{T_η(\mathbb K)}\,dμ(η). $$ Integrating by parts with respect to the measure $μ$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\ge2$, there exists a conservative diffusion process on $\mathbb K(\mathbb R^d)$ which is properly associated with the Dirichlet form $\mathscr E^{\mathbb K}$.

math.PR↗