arXiv subjects

Arnaud Le Ny

Publications and source records attributed to Arnaud Le Ny.

At least 19 recordsLinked to original sources

arXiv2026

Delocalisation and scaling limit for the disordered long-range Discrete Gaussian Chain

In this paper, we study the effect of an external random field on the localisation / delocalisation of the long-range Discrete Gaussian Chain (DGC), recently studied by Garban arXiv:2312.04536 and Coquille \emph{et al.} arXiv:2412.15782. We prove that the model is delocalised at every inverse temperature for any polynomial decay $α> 3/2$. This behaviour differs qualitatively from the one of the disorder-free long-range DGC, for which delocalisation at every temperature occurs only for $α> 2$, similarly to what happens for long-range Ising models as already proved by Aizenman and Wehr. Additionally, we provide a more detailed description of the delocalised phase by identifying the growth exponent and scaling limit of the height at the origin for any decay exponent $α> 3/2$.

math.PR↗

arXiv2026

Quantitative delocalisation for the Gaussian and $q$-SOS long-range chains

The goal of this article is to study quantitatively the localisation/delocalisation properties of the discrete Gaussian chain with long-range interactions. Specifically, we consider the discrete Gaussian chain of length $N$, with Dirichlet boundary condition, range exponent $α\in (1 , \infty)$ and inverse temperature $β\in (0,\infty)$, and show that: - For $α\in (2 ,3)$ and $β\in (0 , \infty)$, the fluctuations of the chain are at least of order $N^{\frac{1}{2}(α- 2)}$; - For $α= 3$ and $β\in (0 , \infty)$, the fluctuations of the chain are of order $\sqrt{N / \ln N}$ (sharp upper and lower bounds up to multiplicative constants are derived). Combined with the results of Kjaer-Hilhorst, Fröhlich-Zegarlinski and Garban, these estimates provide an (almost) complete picture for the localisation/delocalisation of the discrete Gaussian chain. The proofs are based on graph surgery techniques which have been recently developed by van Engelenburg-Lis and Aizenman-Harel-Peled-Shapiro to study the phase transitions of two dimensional integer-valued height functions (and of their dual spin systems). Additionally, by combining the previous strategy with a technique introduced by Sellke, we are able extend the method to study the $q$-SOS long-range chain with exponent $q \in (0 , 2)$ and show that, for any inverse temperature $β\in (0, \infty)$ and any range exponent $α\in (1 , \infty)$: - The fluctuations of the chain are at least of order $N^{\frac{1}{q}(α-2) \wedge \frac{1}{2}}$; - The fluctuations of the chain are at most of order $N^{\left( \frac{1}{q}α- 1 \right) \wedge \frac 12}$.

math.PR↗

arXiv2025

Extremal Ising Gibbs States on Lobachevsky lattices

We exhibit an uncountable family of extremal inhomogeneous Gibbs measures of the low temperature Ising model on regular tilings of the hyperbolic plane. These states arise as low temperature perturbations of local ground states having a sparse enough set of frustrated edges, the sparseness being measured in terms of the isoperimetric constant of the graph. This result is implied by an extension of the article [5] on regular trees to non-amenable graphs. We moreover argue how we can deduce the extremality of an uncountable subset of the Series--Sinai states [23] at low temperature.

math.PR↗

arXiv2024

Absence of shift-invariant Gibbs states (delocalisation) for one-dimensional $\mathbb Z$-valued fields with Long-Range interactions

We show that a modification of the proof of our paper [CvELNR18], in the spirit of [FP81], shows delocalisation in the long-range Discrete Gaussian Chain, and generalisations thereof, for any decay power $α>2$ and at all temperatures. The argument proceeds by contradiction: any shift-invariant and localised measure (in the $L^1$ sense), is a convex combination of ergodic localised measures. But the latter cannot exist: on one hand, by the ergodic theorem, the average of the field over growing boxes would be almost surely bounded ; on the other hand the measure would be absolutely continuous with respect to its height-shifted translates, as a simple relative entropy computation shows. This leads to a contradiction and answers, in a non-quantitative way, an open question stated in a recent paper [G23] of C.Garban.

math.PR↗

arXiv2024

On Long Range Ising Models with Random Boundary Conditions

We consider polynomial long-range Ising models in one dimension, with ferromagnetic pair interactions decaying with power $2-α$ (for $0 \leq α< 1$), and prepared with randomly chosen boundary conditions. We show that at low temperatures in the thermodynamic limit the finite-volume Gibbs measures do not converge, but have a distributional limit, the so-called metastate. We find that there is a distinction between the values of $α$ less than or larger than $\frac{1}{2}$. For moderate, or intermediate, decay $α< \frac{1}{2}$, the metastate is very dispersed and supported on the set of all Gibbs measures, both extremal and non-extremal, whereas for slow decays $α> \frac{1}{2}$ the metastate is still dispersed, but has its support just on the set of the two extremal Gibbs measures, the plus measure and the minus measure. The former, moderate decays case, appears to be new and is due to the occurrence of almost sure boundedness of the random variable which is the sum of all interaction (free) energies between random and ordered half-lines, when the decay is fast enough, but still slow enough to get a phase transition ($α>0$); while the latter, slow decays case, is more reminiscent of and similar to the behaviour of higher-dimensional nearest-neighbour Ising models with diverging boundary (free) energies. We leave the threshold case $α=\frac{1}{2}$ for further studies.

math-ph↗

arXiv2023

Continuity of the extremal decomposition of the free state for finite-spin models on Cayley trees

We prove the continuity of the extremal decomposition measure of the free state of low temperature Potts models, and more generally of ferromagnetic finite-spin models, on a regular tree, including general clock models. The decomposition is supported on uncountably many inhomogeneous extremal states, that we call glassy states. The method of proof provides explicit concentration bounds on branch overlaps, which play the role of an order parameter for typical extremals. The result extends to the counterpart of the free state (called central state) in a wide range of models which have no symmetry, allowing also the presence of sufficiently small field terms. Our work shows in particular that the decomposition of central states into uncountably many glassy states in finite-spin models on trees at low temperature is a generic phenomenon, and does not rely on symmetries of the Hamiltonian.

math.PR↗

arXiv2022

Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees

We consider $\mathbb Z$-valued $p$-SOS-models with nearest neighbor interactions of the form $|ω_v-ω_w|^p$, and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any symmetries of the tree. This generalises the results of Gandolfo, Ruiz and Shlosman \cite{GRS12} about the Ising model, and shows that the latter behaviour is robust. We treat three different types of extensions: non-compact state space gradient models, models without spin-symmetry, and models in small random fields. We give a detailed construction and full proofs of the extremality of the low-temperature states in the set of all Gibbs measures, analysing excess energies relative to the ground states, convergence of low-temperature expansions, and properties of cutsets.

math.PR↗

arXiv2022

Decimations for One- and Two-dimensional Ising and Rotator Models II: Continuous versus Discrete Symmetries

We show how decimated Gibbs measures which have an unbroken continuous symmetry due to the Mermin-Wagner theorem, although their discrete equivalents have a phase transition, still can become non-Gibbsian. The mechanism rests on the occurrence of a spin-flop transition with a broken discrete symmetry, once the model is constrained by the decimated spins in a suitably chosen "bad" configuration.

math-ph↗

arXiv2022

Almost Gibbsian Measures on a Cayley Tree

We consider the ferromagnetic n.n Ising model on Cayley trees in absence of external fields submitted to a modified majority rule transformation with overlapping cells already known to lead to non-Gibbsian measures. We describe the renormalized measures within the Generalized Gibbs framework and prove that they are Almost Gibbs at any temperature.

math-ph↗

arXiv2021

Decimations for Two-dimensional Ising and Rotator Models I

We extend proofs of non-Gibbsianness of decimated Gibbs measures at low temperatures to include long-range, as well as vector-spin interactions. Our main tools consist in a two-dimensional use of ``Equivalence of boundary conditions'' in the long-range case and an extension of Global specifications for two-dimensional vector spins.

math-ph↗

arXiv2020

Markov and almost Markov properties in one, two or more directions

In this review-type paper written at the occasion of the Oberwolfach workshop {\em One-sided vs. Two-sided stochastic processes} (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we discuss in both contexts various extensions of Markov chains and Markov fields and their properties, such as $g$-measures, Variable Length Markov Chains, Variable Neighbourhood Markov Fields, Variable Neighbourhood (Parsimonious) Random Fields, and Generalized Gibbs Measures.

math.PR↗

arXiv2019

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results.

math-ph↗

arXiv2019

Gibbs Measures for Long-Range Ising Models

This review-type paper is based on a talk given at the conference {É}tats de la Recherche en M{é}canique statistique, which took place at IHP in Paris (December 10-14, 2018). We revisit old results from the 80's about one dimensional long-range polynomially decaying Ising models (often called Dyson models in dimension one) and describe more recent results about interface fluctuations and interface states in dimensions one and two.

math-ph↗

arXiv2018

Absence of Dobrushin states for $2d$ long-range Ising models

We consider the two-dimensional Ising model with long-range pair interactions of the form $J_{xy}\sim|x-y|^{-α}$ with $α>2$, mostly when $J_{xy} \geq 0$. We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed $\pm$-boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman-Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts.

math.PR↗

arXiv2017

Entropic repulsion and lack of the $g$-measure property for Dyson models

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.

math-ph↗

arXiv2017

Decimation of the Dyson-Ising Ferromagnet

We study the decimation to a sublattice of half the sites, of the one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair interactions of the form $\frac{1}{{|i-j|}^α}$, in the phase transition region (1< $α\leq$ 2, and low temperature). We prove non-Gibbsianness of the decimated measure at low enough temperatures by exhibiting a point of essential discontinuity for the finite-volume conditional probabilities of decimated Gibbs measures. Thus result complements previous work proving conservation of Gibbsianness for fastly decaying potentials ($α$ > 2) and provides an example of a "standard" non-Gibbsian result in one dimension, in the vein of similar resuts in higher dimensions for short-range models. We also discuss how these measures could fit within a generalized (almost vs. weak) Gibbsian framework. Moreover we comment on the possibility of similar results for some other transformations.

math-ph↗

arXiv2016

Persistent random walks. II. Functional Scaling Limits

We give a complete and unified description -- under some stability assumptions -- of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the memory. It turns out that the limit process is either Markovian or not according to -- to put it in a nutshell -- the rate of decrease of the distribution tails corresponding to the persistent times. In the memoryless situation, the limits are classical strictly stable L{é}vy processes of infinite variations. However, we point out that the description of the critical Cauchy case fills some lacuna even in the closely related context of Directionally Reinforced Random Walks (DRRWs) for which it has not been considered yet. Besides, we need to introduced some relevant generalized drift -- extended the classical one -- in order to study the critical case but also the situation when the limit is no longer Markovian. It appears to be in full generality a drift in mean for the Persistent Random Walk (PRW). The limit processes keeping some memory -- given by some variable length Markov chain -- of the underlying PRW are called arcsine Lamperti anomalous diffusions due to their marginal distribution which are computed explicitly here. To this end, we make the connection with the governing equations for L{é}vy walks, the occupation times of skew Bessel processes and a more general class modelled on Lamperti processes. We also stress that we clarify some misunderstanding regarding this marginal distribution in the framework of DRRWs. Finally, we stress that the latter situation is more flexible -- as in the first paper -- in the sense that the results can be easily generalized to a wider class of PRWs without renewal pattern.

math.PR↗

arXiv2015

Persistent random walks

We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent random walk and give the conditions of recurrence or transience in terms of "transition" probabilities to keep on the same direction or to change, without assuming that the latter admits any stationary probability. Examples are exhibited when this process is recurrent even if the random walk is not symmetric.

math.PR↗