arXiv subjects

Louis-Pierre Arguin

Publications and source records attributed to Louis-Pierre Arguin.

45 records · Page 3Linked to original sources

arXiv2009

Behavior of the two-dimensional Ising model at the boundary of a half-infinite cylinder

The two-dimensional Ising model is studied at the boundary of a half-infinite cylinder. The three regular lattices (square, triangular and hexagonal) and the three regimes (sub-, super- and critical) are discussed. The probability of having precisely 2n spinflips at the boundary is computed as a function of the positions k_i's, i=1,..., 2n, of the spinflips. The limit when the mesh goes to zero is obtained. For the square lattice, the probability of having 2n spinflips, independently of their position, is also computed. As a byproduct we recover a result of De Coninck showing that the limiting distribution of the number of spinflips is Gaussian. The results are obtained as consequences of Onsager's solution and are rigorous.

cond-mat.stat-mech↗

arXiv2009

On the structure of quasi-stationary competing particle systems

We study point processes on the real line whose configurations $X$ are locally finite, have a maximum and evolve through increments which are functions of correlated Gaussian variables. The correlations are intrinsic to the points and quantified by a matrix $Q=\{q_{ij}\}_{i,j\in\mathbb{N}}$. A probability measure on the pair $(X,Q)$ is said to be quasi-stationary if the joint law of the gaps of $X$ and of $Q$ is invariant under the evolution. A known class of universally quasi-stationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchically nested Poisson--Dirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi-stationary. The main result of this work is a proof of this conjecture for the case where $q_{ij}$ assume only a finite number of values. The result is of relevance for mean-field spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchical organization of the Gibbs measure was first proposed as an ansatz.

math.PR↗

arXiv2009

On Small Perturbations of a Spin Glass System

We show through a simple example that perturbations of the Hamiltonian of a spin glass which cannot be detected at the level of the free energy can completely alter the behavior of the overlap. In particular, perturbations of order O(log N), with N the size of the system, suffice to have ultrametricity emerge in the thermodynamical limit.

math-ph↗

arXiv2008

A Remark on the Infinite-Volume Gibbs Measures of Spin Glasses

In this note, we point out that infinite-volume Gibbs measures of spin glass models on the hypercube can be identified as random probability measures on the unit ball of a Hilbert space. This simple observation follows from a result of Dovbysh and Sudakov on weakly exchangeable random matrices. Limiting Gibbs measures can then be studied as single well-defined objects. This approach naturally extends the space of Random Overlap Structures as defined by Aizenman, Sims and Starr. We discuss the Ruelle Probability Cascades and the stochastic stability within this framework. As an application, we use an idea of Parisi and Talagrand to prove that if a sequence of finite-volume Gibbs measures satisfies the Ghirlanda-Guerra identities, then the infinite-volume measure must be singular as a measure on a Hilbert space.

math.PR↗

arXiv2007

Competing Particle Systems and the Ghirlanda-Guerra Identities

We study point processes on the real line whose configurations X can be ordered decreasingly and evolve by increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q={q_ij}. Quasi-stationary systems are those for which the law of (X,Q) is invariant under the evolution up to translation of X. It was conjectured by Aizenman and co-authors that the matrix Q of robustly quasi-stationary systems must exhibit a hierarchal structure. This was established recently, up to a natural decomposition of the system, whenever the set S_Q of values assumed by q_ij is finite. In this paper, we study the general case where S_Q may be infinite. Using the past increments of the evolution, we show that the law of robustly quasi-stationary systems must obey the Ghirlanda-Guerra identities, which first appear in the study of spin glass models. This provides strong evidence that the above conjecture also holds in the general case.

math.PR↗

arXiv2007

A dynamical characterization of Poisson-Dirichlet distributions

We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(a,0)$. Precisely, let $s$ be a proper random mass-partition i.e. a random sequence $(s_i, i\in\N)$ such that $s_1\geq s_2\geq ...$ and $\sum_i s_i=1$ a.s. Consider ${W_i}_{i\in\N}$, an iid sequence of positive random variables with a density and such that $E[W^λ]$ is finite for all $λ\in\R$. It is shown that if the law of $s$ is invariant under a random multiplicative shift $s_i W_i$ of the atoms followed by a renormalization, then it must be a mixture of Poisson-Dirichlet distribution $PD(a,0)$, $a\in (0,1)$.

math.PR↗

arXiv2006

Spin Glass Computations and Ruelle's Probability Cascades

We study the Parisi functional, appearing in the Parisi formula for the pressure of the SK model, as a functional on Ruelle's Probability Cascades (RPC). Computation techniques for the RPC formulation of the functional are developed. They are used to derive continuity and monotonicity properties of the functional retrieving a theorem of Guerra. We also detail the connection between the Aizenman-Sims-Starr variational principle and the Parisi formula. As a final application of the techniques, we rederive the Almeida-Thouless line in the spirit of Toninelli but relying on the RPC structure.

math-ph↗

arXiv2002

Homology of Fortuin-Kasteleyn clusters of Potts models on the torus

Topological properties of Fortuin-Kasteleyn clusters are studied on the torus. Namely, the probability that their topology yields a given subgroup of the first homology group of the torus is computed for Q=1, 2, 3 and 4. The expressions generalize those obtained by Pinson for percolation (Q=1). Numerical results are also presented for three tori of different moduli. They agree with the theoretical predictions for Q=1, 2 and 3. For Q=4 agreement is not ruled out but what seems logarithmic corrections makes it harder to decide.

hep-th↗

arXiv2002

Non-unitary observables in the 2d critical Ising model

We introduce three non-local observables for the two-dimensional Ising model. At criticality, conformal field theory may be used to obtain theoretical predictions for their behavior. These formulae are explicit enough to show that their asymptotics are described by highest weights $h_{pq}$ from the Kac table for c=1/2 distinct from those of the three unitary representations (0, 1/16 and 1/2).

hep-th↗