arXiv subjects

Yvan Velenik

Publications and source records attributed to Yvan Velenik.

At least 19 recordsLinked to original sources

arXiv2026

Typical geometry of self-repelling polymers in a constant force field

We study a general class of self-repelling polymers on $\mathbb Z^2$, including the simple random walk, the self-avoiding walk and the repulsive Domb-Joyce model, in the presence of a constant force field acting on each monomer. Conditioning the polymer to have fixed length and fixed endpoints, we identify the limiting free energy and prove that typical trajectories concentrate exponentially near a deterministic macroscopic shape. This shape is characterized as the unique minimizer of a variational problem and can be interpreted as a geodesic of a height-dependent Finsler metric. We also analyze two limiting regimes with universal features: for small field strength, in the symmetric case, the geodesic is close to a classical catenary, while for large field strength it converges to a universal polygonal shape governed by the nearest-neighbor lattice constraint.

math-ph↗

arXiv2025

Finite-time trajectorial estimates for inhomogeneous random walks

We consider integer-valued random walks with independent but not identically distributed increments, and extend to this context several classical estimates, including a local limit theorem, precise small-ball estimates (both conditional on the final point and unconditional), and bounds on the probability that the random walk trajectory remains positive up to a given time (again, both conditional on the final point and unconditional). Two key features of this work are that the bounds are non-asymptotic, holding true for finite time horizons, and, crucially, that the latter hold uniformly over an entire class of admissible increment sequences. This provides a robust framework for applications. These results are, in particular, tailored for the analysis of processes derived through a time-dependent tilting of the increments of a time-homogeneous random walk.

math.PR↗

arXiv2025

Random Walk conditioned to stay above a non-flat floor: curvature effects

Let $h:[0,1]\to\mathbb{R}$ be $C^2$ and such that $\sup_{[0,1]} h''<0$. For a (large) positive integer $n$, set $h_n(k) = n h(k/n)$ for any $k\in\{0,\dots,n\}$. We consider a random walk $(S_k)_{k\geq 0}$ with i.i.d.\ centred increments having some finite exponential moments. We are interested in the event $\{S\geq h_n\} = \{S_k\geq h_n(k)\;\forall k\in\{0,\dots,n\}\}$. It is well known that $P(S\geq h_n \,|\, S_0=0,\, S_n=\lceil h_n(n) \rceil) = e^{n\int_0^1 I(h'(s)) \,ds + o(n)}$, where $I$ is the Legendre-Fenchel transform of the log-moment generating function associated to the increments. We first prove that the leading correction is of order $e^{-Θ(n^{1/3})}$. We then turn our attention to the conditional random walk measure $P^h_n = P(\cdot \,|\, S\geq h_n, S_0=0, S_n=\lceil h_n(n) \rceil)$. We prove that the one-point tails are of the form $\mathbb{P}_n^h (S_k \geq h_n(k) + t n^{1/3} ) = e^{-Θ(t^{3/2})}$ for all $t<n^β$ for any $β\in (0,1/6)$. Moreover, we prove that, for any $r\geq 1$, $E_n^h((S_k-h_n(k))^r) = Θ(n^{r/3})$ and $\mathrm{Var}_{P_n^h}(S_k) = Θ(n^{2/3})$, for all $k$ far enough from $0$ and $n$. In addition, we show that $\mathrm{Cov}_{P_n^h}(S_k,S_\ell) \leq e^{-O(|\ell-k|/n^{2/3})}$ for all $k,\ell$ not too close to $0$ and $n$.

math.PR↗

arXiv2024

Fixed-magnetization Ising model with a slowly varying magnetic field

The motivation for this paper is the analysis of the fixed-density Ising lattice gas in the presence of a gravitational field. This is a seen as a particular instance of an Ising model with a slowly varying magnetic field in the fixed magnetization ensemble. We first characterize the typical magnetization profiles in the regime in which the contribution of the magnetic field competes with the bulk energy term. We then discuss in more detail the particular case of a gravitational field and the arising interfacial phenomena. In particular, we identify the macroscopic profile and propose several conjectures concerning the interface appearing in the phase coexistence regime. The latter are supported by explicit computations in an effective model. Finally, we state some conjectures concerning equilibrium crystal shapes in the presence of a gravitational field, when the latter contributes to the energy only to surface order.

math.PR↗

arXiv2023

Asymptotics of correlations in the Ising model: a brief survey

We present a brief survey of rigorous results on the asymptotic behavior of correlations between two local functions as the distance between their support diverges, concentrating on the Ising model on $\mathbb{Z}^d$ with finite-range ferromagnetic interactions.

math.PR↗

arXiv2022

On the two-point function of the Potts model in the saturation regime

We consider the Random-Cluster model on $\mathbb{Z}^d$ with interactions of infinite range of the form $J_x = ψ(x)\mathsf{e}^{-ρ(x)}$ with $ρ$ a norm on $\mathbb{Z}^d$ and $ψ$ a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of $β_{\rm sat}(s)>0$ such that the inverse correlation length in the direction $s$ is constant on $[0,β_{\rm sat}(s))$), thus removing a regularity assumption used in a previous work of ours. Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein-Zernike form) for the two-point function in the whole saturation regime $(0,β_{\rm sat}(s))$. We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in $β$ in the regime $(β_{\rm sat}(s), β_{\rm c})$.

math.PR↗

arXiv2022

Ornstein-Zernike behavior for Ising models with infinite-range interactions

We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions. Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk. We aimed at a pedagogical and self-contained presentation.

math-ph↗

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

Asymptotics of even-even correlations in the Ising model

We consider finite-range ferromagnetic Ising models on $\mathbb{Z}^d$ in the regime $β<β_c$. We analyze the behavior of the prefactor to the exponential decay of $\mathrm{Cov}(σ_A,σ_B)$, for arbitrary finite sets $A$ and $B$ of even cardinality, as the distance between $A$ and $B$ diverges.

math.PR↗

arXiv2018

Potts models with a defect line

We provide a detailed analysis of the correlation length in the direction parallel to a line of modified coupling constants in the ferromagnetic Potts model on $\mathbb{Z}^d$ at temperatures $T>T_c$. We also describe how a line of weakened bonds pins the interface of the Potts model on $\mathbb{Z}^2$ below its critical temperature. These results are obtained by extending the analysis by Friedli, Ioffe and Velenik from Bernoulli percolation to FK-percolation of arbitrary parameter $q>1$.

math-ph↗

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↗

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↗

arXiv2013

Upper bound on the decay of correlations in a general class of O(N)-symmetric models

We consider a general class of two-dimensional spin systems, with continuous but not necessarily smooth, possibly long-range, $O(N)$-symmetric interactions, for which we establish algebraically decaying upper bounds on spin-spin correlations under all infinite-volume Gibbs measures. As a by-product, we also obtain estimates on the effective resistance of a (possibly long-range) resistor network in which randomly selected edges are shorted.

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↗