arXiv subjects

Pierre-Loïc Méliot

Publications and source records attributed to Pierre-Loïc Méliot.

At least 19 recordsLinked to original sources

arXiv2024

A determinantal point process approach to scaling and local limits of random Young tableaux

We obtain scaling and local limit results for large random Young tableaux of fixed shape $λ^0$ via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). More precisely, we prove: (1) an explicit description of the limiting surface of a uniform random Young tableau of shape $λ^0$, based on solving a complex-valued polynomial equation; (2) a simple criteria to determine if the limiting surface is continuous in the whole domain; (3) and a local limit result in the bulk of a random Poissonized Young tableau of shape $λ^0$. Our results have several consequences, for instance: they lead to explicit formulas for the limiting surface of $L$-shaped tableaux, generalizing the results of Pittel and Romik (2007) for rectangular shapes; they imply that the limiting surface for $L$-shaped tableaux is discontinuous for almost-every $L$-shape; and they give a new one-parameter family of infinite random Young tableaux, constructed from the so-called random infinite bead process.

math.PR↗

arXiv2022

Mod-Poisson approximation schemes: Applications to credit risk

We introduce a new numerical approximation method for functionals of factor credit portfolio models based on the theory of mod-$ϕ$ convergence and mod-$ϕ$ approximation schemes. The method can be understood as providing correction terms to the classic Poisson approximation, where higher order corrections lead to asymptotically better approximations as the number of obligors increases. We test the model empirically on two tasks: the estimation of risk measures ($\mathrm{VaR}$ and $\mathrm{ES}$) and the computation of CDO tranche prices. We compare it to other commonly used methods -- such as the recursive method, the large deviations approximation, the Chen--Stein method and the Monte Carlo simulation technique (with and without importance sampling) -- and we show that it leads to more accurate estimates while requiring less computational time.

q-fin.CP↗

arXiv2022

Mod-Poisson approximation schemes and higher-order Chen-Stein inequalities

In this article, we provide an extension of the Chen-Stein inequality for Poisson approximation in the total variation distance for sums of independent Bernoulli random variables in two ways. We prove that we can improve the rate of convergence (hence the quality of the approximation) by using explicitly constructed signed or positive probability measures, and that we can extend the setting to possibly dependent random variables. The framework which allows this is that of mod-Poisson convergence, and more precisely those mod-Poisson convergent sequences whose residue functions can be expressed as a specialization of the generating series of elementary symmetric functions. This combinatorial reformulation allows us to have a general and unified framework in which we can fit the classical setting of sums of independent Bernoulli random variables as well as other examples coming e.g. from probabilistic number theory and random permutations.

math.PR↗

arXiv2022

Asymptotics of the major index of a random standard tableau

In this article, we establish the mod-$ϕ$ convergence of the major index of a uniform random standard tableau whose shape converges in the Thoma simplex. This implies various probabilistic estimates, in particular speed of convergence estimates of Berry--Esseen type, and strong large deviation principles.

math.PR↗

arXiv2022

Gaussian approximations for random vectors

We present several refinements on the fluctuations of sequences of random vectors (with values in the Euclidean space $\mathbb{R}^d$) which converge after normalization to a multidimensional Gaussian distribution. More precisely we refine such results in two directions: first we give conditions under which one can obtain bounds on the speed of convergence to the multidimensional Gaussian distribution, and then we provide a setting in which one can obtain precise moderate or large deviations (in particular we see at which scale the Gaussian approximation for the tails ceases to hold and how the symmetry of the Gaussian tails is then broken). These results extend some of our earlier works obtained for real valued random variables, but they are not simple extensions, as some new phenomena are observed that could not be visible in one dimension. Even for very simple objects such as the symmetric random walk in $\mathbb{Z}^d$, we observe a loss of symmetry that we can quantify for walks conditioned to be far away from the origin. Also, unlike the one dimensional case where the Kolmogorov distance is natural, in the multidimensional case there is no more such a canonical distance. We choose to work with the so-called convex distance, and as a consequence, the geometry of the Borel measurable sets that we consider shall play an important role (also making the proofs more complicated). We illustrate our results with some examples such as correlated random walks, the characteristic polynomials of random unitary matrices, or pattern countings in random graphs.

math.PR↗

arXiv2022

On the precise deviations of the characteristic polynomial of a random matrix

In this paper, using techniques developed in our earlier works on the theory of mod-Gaussian convergence, we prove precise moderate and large deviation results for the logarithm of the characteristic polynomial of a random unitary matrix. In the case where the unitary matrix is chosen according to the Haar measure, the logarithms of the probabilities of fluctuations of order $A=O(N)$ of the logarithm of the characteristic polynomial have been estimated by Hughes, Keating and O'Connell. In this work we give an equivalent of the probabilities themselves (without the logarithms), and we do so for the more general case of a matrix from the circular $β$ ensemble for any parameter $β> 0$. In comparison to previous results from Féray-Méliot-Nikeghbali (2016) and Dal Borgo-Hovhannisyan-Rouault (2019), we considerably extend the range of fluctuations for which precise estimates can be written.

math.PR↗

arXiv2021

A central limit theorem for singular graphons

We associate to a graphon $γ$ the sequence of $W$-random graphs $(G_n(γ))_{n \geq 1}$. We say that the graphon is singular if, for any finite graph $F$, the homomorphism density $t(F,G_n(γ))$ has a variance of order $O(n^{-2})$. This behavior is singular because generically, the density of a fixed finite graph $F$ in a $W$-random graph has a variance of order $O(n^{-1})$. We conjecture that the only singular graphons are the constant graphons $γ_p$ with $p \in [0,1]$, corresponding to the Erdős-Rényi random graphs $G(n,p)$. In this paper, we investigate the general properties of the singular graphons, and we show that they share many properties with the Erdős-Rényi random graphs. In particular, if $γ$ is a singular graphon, then the scaled densities $n(t(F,G_n(γ))-\mathbb{E}[t(F,G_n(γ))])$ converge in joint distribution. This generalises the central limit theorem satisfied by the Erdős-Rényi random graphs $G(n,p)$; however, the limiting distribution might be non-Gaussian if the conjecture does not hold. We also establish an equation satisfied by the characteristic polynomial of the Laplacian of the graph $G_n(γ)$ associated to a singular graphon; this opens the way to a spectral approach of the conjecture.

math.PR↗

arXiv2021

Fluctuations of the Gromov-Prohorov sample model

In this paper, we study the fluctuations of observables of metric measure spaces which are random discrete approximations $X_n$ of a fixed arbitrary (complete, separable) metric measure space $X=(\mathcal{X},d,μ)$. These observables $Φ(X_n)$ are polynomials in the sense of Greven-Pfaffelhuber-Winter, and we show that for a generic model space $X$, they yield asymptotically normal random variables. However, if $X$ is a compact homogeneous space, then the fluctuations of the observables are much smaller, and after an adequate rescaling, they converge towards probability distributions which are not Gaussian. Conversely, we prove that if all the fluctuations of the observables $Φ(X_n)$ are smaller than in the generic case, then the measure metric space $X$ is compact homogeneous. The proofs of these results rely on the Gromov reconstruction principle, and on an adaptation of the method of cumulants and mod-Gaussian convergence developed by Féray-Méliot-Nikeghbali.

math.PR↗

arXiv2019

Local limit theorems and mod-phi convergence

We prove local limit theorems for mod-ϕ convergent sequences of random variables, ϕ being a stable distribution. In particular, we give two new proofs of a local limit theorem in the framework of mod-phi convergence: one proof based on the notion of zone of control, and one proof based on the notion of mod-ϕ convergence in L1(iR). These new approaches allow us to identify the infinitesimal scales at which the stable approximation is valid. We complete our analysis with a large variety of examples to which our results apply, and which stem from random matrix theory, number theory, combinatorics or statistical mechanics.

math.PR↗

arXiv2018

Asymptotic representation theory and the spectrum of a random geometric graph on a compact Lie group

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $Γ(N,L)$ whose vertices are $N$ random points $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $Γ(N,L)$, when $N$ goes to infinity. If $L$ is fixed and $N \to + \infty$ (Gaussian regime), then the largest eigenvalues of $Γ(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions. If $L = O(N^{-\frac{1}{\dim G}})$ and $N \to +\infty$ (Poissonian regime), then the random geometric graph $Γ(N,L)$ converges in the local Benjamini-Schramm sense, which implies the weak convergence in probability of the spectral measure of $Γ(N,L)$. In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl's character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.

math.PR↗

arXiv2018

Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces

In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space.

math.PR↗

arXiv2018

Mod-$ϕ$ convergence, II: Estimates on the speed of convergence

In this paper, we give estimates for the speed of convergence towards a limiting stable law in the recently introduced setting of mod-$ϕ$ convergence. Namely, we define a notion of zone of control, closely related to mod-$ϕ$ convergence, and we prove estimates of Berry-Esseen type under this hypothesis. Applications include: the winding number of a planar Brownian motion; classical approximations of stable laws by compound Poisson laws; examples stemming from determinantal point processes (characteristic polynomials of random matrices and zeroes of random analytic functions); sums of variables with an underlying dependency graph (for which we recover a result of Rinott, obtained by Stein's method); the magnetization in the $d$-dimensional Ising model; and functionals of Markov chains.

math.PR↗

arXiv2017

Mod-$ϕ$ convergence: Approximation of discrete measures and harmonic analysis on the torus

In this paper, we relate the framework of mod-$ϕ$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents in the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in (possibly coloured) permutations, number of prime divisors (possibly within different residue classes) of a random integer, number of irreducible factors of a random polynomial, etc. One advantage of the approach developed in this paper is that it allows us to deal with approximations in higher dimensions as well. In this setting, we can explicitly see the influence of the correlations between the components of the random vectors in our asymptotic formulas.

math.PR↗

arXiv2015

Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$ϕ$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erdős-Rényi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.

math.PR↗

arXiv2014

Mod-Gaussian convergence and its applications for models of statistical mechanics

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of "breaking of symmetry" in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of $L^1$-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.

math.PR↗

arXiv2013

Partial isomorphisms over finite fields

In this paper, we construct a combinatorial algebra of partial isomorphisms that gives rise to a "projective limit" of the centers of the group algebras C[GL(n,Fq)]. It allows us to prove a GL(n,Fq)-analogue of an old theorem of Farahat and Higman regarding products of conjugacy classes of permutations.

math.CO↗

arXiv2013

The cut-off phenomenon for Brownian motions on symmetric spaces of compact type

We prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say: (1) the classical simple compact Lie groups: special orthogonal groups, special unitary groups and compact symplectic groups; (2) the real, complex and quaternionic Grassmannian varieties (including the real spheres and complex or quaternionic projective spaces); (3) the spaces of structures: SU(n)/SO(n), SO(2n)/U(n), SU(2n)/USp(n), and USp(n)/U(n). In each case, we give explicit lower bounds for the total variation distance DTV(mu_t,Haar) if t < tcut-off = a log n, and explicit upper bounds if t > tcut-off. This gives in particular an answer to a question raised in recent papers by Chen and Saloff-Coste.

math.PR↗

arXiv2011

A central limit theorem for the characters of the infinite symmetric group and of the infinite Hecke algebra

In this paper, we review the representation theory of the infinite symmetric group, and we extend the works of Kerov and Vershik by proving that the irreducible characters of the infinite symmetric group always satisfy a central limit theorem. Hence, for any point of the Thoma simplex, the corresponding measures on the levels of the Young graph have a property of gaussian concentration. By using the Robinson-Schensted-Knuth algorithm and the theory of Pitman operators, we relate these results to the properties of certain random permutations obtained by riffle shuffles, and to the behaviour of random walks conditioned to stay in a Weyl chamber.

math.RT↗