arXiv subjects

Yacine Barhoumi-Andréani

Publications and source records attributed to Yacine Barhoumi-Andréani.

10 recordsLinked to original sources

arXiv2025

Decoupling the i.i.d. field and the randomisation field in the Curie-Weiss model

Using the De Finetti representation of the Curie-Weiss model, the uniform coupling of Bernoulli random variables and the Laplace inversion formula (almost surely), we show that the full phase diagram of the Curie-Weiss model can be explained by a competition between the De Finetti randomisation and an approximate Gaussian process indexed by a complex variable that is equal to the inverse Laplace transform on a complex line of a Brownian Bridge. A more refined process type of rescaling shows that this is a modification of the Brownian Sheet that is at the core of all Gaussian random variables in the limits obtained in the model. This almost sure Laplace inversion approach allows moreover to treat all types of spin laws in the same vein as the Curie-Weiss Bernoulli spins. This gives a natural explanation of several results that already appeared in the literature in the subcritical and critical case in addition to produce new analogous results in the super-critical case. The functional approach here defined can moreover be extended to a wide class of statistical mechanical models that includes the Ising model in any dimension.

math.PR↗

arXiv2025

Splitting phenomenon in the Sathe-Selberg theorem, mod-Poisson convergence with auxiliary randomisation and universality of the Gamma factor

We consider several sequences of random variables whose Fourier-Laplace transforms present the same type of \textit{splitting phenomenon} when suitably rescaled by the Fourier-Laplace transform of a Poisson-distributed random variable (mod-Poisson convergence). Addressing a question raised by Kowalski-Nikeghbali, we explain the appearance of a universal term, the \textit{Gamma factor}, by a common feature of each model, the existence of an auxiliary randomisation that reveals an independence structure. This universal Gamma factor is a consequence of the probabilistic (exponential) fluctuations of this subordinated random variable. The class of examples that belong to this framework includes: random uniform permutations, random Gauss integers, random polynomials over a finite field, random matrices with values in a finite field, random partitions and the classical Sathe-Selberg theorems in probabilistic number theory. For this last example, the randomisation that triggers the Gamma factor defines a new probability distribution, the ``delta-Zeta distribution'' which is the analogue of the geometric distribution in the permutation setting.

math.PR↗

arXiv2024

A surrogate by exchangeability approach to the Curie-Weiss model

We introduce a new general concept of surrogate random variable, the ``surrogate by exchangeability'' that allows to study the class of random variables that can be decomposed by means of an independent randomisation. As an example, we treat the case of the Curie-Weiss model using the explicit construction of its De Finetti measure of exchangeability. Writing the magnetisation as a sum of i.i.d.'s randomised by the underlying De Finetti random variable, the surrogate study shows that the appearance of a phase transition can be understood as a competition between these two sources of randomness, the Gaussian regime corresponding to a marginally relevant disordered system.

math.PR↗

arXiv2024

Remarkable structures in integrable probability, I: max-independence structures

We analyse in a systematic way the occurrences of a remarkable structure in the theory of integrable probability that we call a ``max-independence structure'', when random variables are constructed as a maximum of a sequence of independent random variables. The list of treated examples contains~: the $ GU\!E $ and $ GO\!E $ Tracy-Widom distributions, the extreme eigenvalues/eigenangles of random hermitian/unitary matrices (and in particular the historical example of the $ GU\!E $ extreme eigenvalues), the Hopf-Cole solution to the $ KPZ $ equation with Dirac initial condition (continuum random polymer) and the symmetric Schur measure. % In this last case, the largest part of the underlying random partition is the maximum of an i.i.d randomisation of the deterministic sequence of negative integers. In the case of the $ GU\!E $ Tracy-Widom distribution, the laws of the random variables use a rescaling of the prolate hyperspheroidal wave functions that were introduced by Heurtley and Slepian in the context of circular optical mirrors, and in the case of the $ GU\!E $, the distributions use the sigma-form of the Painlevé $ I\!V $ equation. To illustrate the utility of such a structure, we rescale the largest eigenvalue of the $ GU\!E $ written as a maximum of $N$ independent random variables with the classical Poisson approximation for sums of indicators. We use for this the Okamoto-Noumi-Yamada theory of the sigma-form of the Painlevé equation applied to random matrix theory by Forrester-Witte. By doing so, we find a new expression for the cumulative distribution function of the $ GU\!E $ Tracy-Widom distribution which is shown to be equivalent to the classical one using manipulations à la Forrester-Witte.

math.PR↗

arXiv2020

A new approach to the characteristic polynomial of a random unitary matrix

Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ \bullet $ its value in $1$ (Keating-Snaith theorem), $ \bullet $ the truncation of its Fourier series up to any fraction of its degree, $ \bullet $ the computation of the relative volume of the Birkhoff polytope, $ \bullet $ its products and ratios taken in different points, $ \bullet $ the product of its iterated derivatives in different points, $ \bullet $ functionals in relation with sums of divisor functions in $ \mathbb{F}_q[X] $. $ \bullet $ its mid-secular coefficients, $ \bullet $ the "moments of moments", etc. We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional.

math.PR↗

arXiv2019

Bivariate fluctuations for the number of arithmetic progressions in random sets

We study arithmetic progressions $\{a,a+b,a+2b,\dots,a+(\ell-1) b\}$, with $\ell\ge 3$, in random subsets of the initial segment of natural numbers $[n]:=\{1,2,\dots, n\}$. Given $p\in[0,1]$ we denote by $[n]_p$ the random subset of $[n]$ which includes every number with probability $p$, independently of one another. The focus lies on sparse random subsets, i.e.\ when $p=p(n)=o(1)$ as $n\to+\infty$. Let $X_\ell$ denote the number of distinct arithmetic progressions of length $\ell$ which are contained in $[n]_p$. We determine the limiting distribution for $X_\ell$ not only for fixed $\ell\ge 3$ but also when $\ell=\ell(n)\to+\infty$. The main result concerns the joint distribution of the pair $(X_{\ell},X_{\ell'})$, $\ell>\ell'$, for which we prove a bivariate central limit theorem for a wide range of $p$. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or non-trivial is characterised by the asymptotic behaviour (as $n\to+\infty$) of the threshold function $ψ_\ell=ψ_\ell(n):=np^{\ell-1}\ell$. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

math.PR↗

arXiv2017

Extreme value statistics for the roots of a complex Kac polynomial

We investigate the fluctuations and large deviations of the root of largest modulus in a model of random polynomial with independent complex Gaussian coefficients (Kac polynomials). The fluctuations were recently computed by R. Butez (arxiv 1704.02761) and involve a Fredholm determinant. The precise large deviations show a particular function defined by a series of mutiple integrals in the same vein.

math.PR↗

arXiv2017

Random characters under the $L$-measure, I : Dirichlet characters

We define the $L$-measure on the set of Dirichlet characters as an analogue of the Plancherel measure, once considered as a measure on the irreducible characters of the symmetric group. We compare the two measures and study the limit in distribution of characters evaluations when the size of the underlying group grows. These evaluations are proven to converge in law to imaginary exponentials of a Cauchy distribution in the same way as the rescaled windings of the complex Brownian motion. This contrasts with the case of the symmetric group where the renormalised characters converge in law to Gaussians after rescaling (Kerov Central Limit Theorem).

math.PR↗

arXiv2017

A penalised model reproducing the mod-Poisson fluctuations in the Sathe-Selberg theorem

We construct a probabilistic model for the number of divisors of a random uniform integer that converges in the mod-Poisson sense to the same limiting function as its original counterpart, the one arising in the Sathe-Selberg theorem. This construction involves a conditioning and gives an alternative perspective to the usual paradigm of "hybrid product" models developed by Gonek, Hughes and Keating in the case of the Riemann Zeta function.

math.PR↗

arXiv2017

On Stein's method and mod-* convergence

Stein's method allows to prove distributional convergence of a sequence of random variables and to quantify it with respect to a given metric such as Kolmogorov's (a Berry-Esséen type theorem). Mod-* convergence quantifies the convergence of a sequence of random variables to a given distribution in a sense unusual in probability theory, a priori unrelated to a metric on probability measures. This article gives a connection between these two notions. It shows that mod-* convergence can be understood as a higher order approximation in distribution when the limiting function is integrable and proves a refined Berry-Esséen type theorem for sequences converging in the mod-Gaussian sense.

math.PR↗