arXiv subjects

Ronan Herry

Publications and source records attributed to Ronan Herry.

15 recordsLinked to original sources

arXiv2025

Gradient flow of the infinite-volume free energy for lattice systems of continuous spins

We consider an infinite lattice system of interacting spins living on a smooth compact manifold, with short- but not necessarily finite-range pairwise interactions. We construct the gradient flow of the infinite-volume free energy on the space of translation-invariant spin measures, using an adaptation of the variational approach in Wasserstein space pioneered by Jordan, Kinderlehrer, and Otto. We also construct the infinite-volume diffusion corresponding to the so-called overdamped Langevin dynamics of the spins under the effect of the interactions and of thermal agitation. We show that the trajectories of the gradient flow and of the law of the spins under this diffusion both satisfy, in a weak sense, the same hierarchy of coupled parabolic PDE's, which we interpret as an infinite-volume Fokker-Planck-Kolmogorov equation. We prove regularity of weak solutions and derive an Evolution Variational Inequality for regular solutions, which implies uniqueness. Thus, in particular, the trajectories of the gradient flow coincide with those obtained from the Langevin dynamics. Concerning the long-time evolution, we check that the free energy is always non-increasing along the flow and that moreover, if the Ricci curvature of the spin space is uniformly positive, then at high enough temperature the dynamics converges exponentially, in free energy and in specific Wasserstein distance, to the unique minimizer of the infinite-volume free energy.

math.PR↗

arXiv2024

Limit distributions for polynomials with independent and identically distributed entries

We characterize the limiting distributions of random variables of the form $P_n\left( (X_i)_{i \ge 1} \right)$, where: (i) $(P_n)_{n \ge 1}$ is a sequence of multivariate polynomials, each potentially involving countably many variables; (ii) there exists a constant $D \ge 1$ such that for all $n \ge 1$, the degree of $P_n$ is bounded above by $D$; (iii) $(X_i)_{i \ge 1}$ is a sequence of independent and identically distributed random variables, each with zero mean, unit variance, and finite moments of all orders. More specifically, we prove that the limiting distributions of these random variables can always be represented as the law of $P_\infty\left( (X_i, G_i)_{i \ge 1} \right)$, where $P_\infty$ is a polynomial of degree at most $D$ (potentially involving countably many variables), and $(G_i)_{i \ge 1}$ is a sequence of independent standard Gaussian random variables, which is independent of $(X_i)_{i \ge 1}$. We solve this problem in full generality, addressing both Gaussian and non-Gaussian inputs, and with no extra assumption on the coefficients of the polynomials. In the Gaussian case, our proof builds upon several original tools of independent interest, including a new criterion for central convergence based on the concept of maximal directional influence. Beyond asymptotic normality, this novel notion also enables us to derive quantitative bounds on the degree of the polynomial representing the limiting law. We further develop techniques regarding asymptotic independence and dimensional reduction. To conclude for polynomials with non-Gaussian inputs, we combine our findings in the Gaussian case with invariance principles.

math.PR↗

arXiv2024

The Brenier-Schr\"odinger problem with respect to Feller semimartingales and non-local Hamilton-Jacobi-Bellman equations

Motivated by a problem from incompressible fluid mechanics of Brenier (JAMS 1989), and its recent entropic relaxation by Arnaudo, Cruizero, L\'eonard & Zambrini (AIHP PS 2020), we study a problem of entropic minimization on the path space when the reference measure is a generic Feller semimartingale. We show that, under some regularity condition, our problem connects naturally with a, possibly non-local, version of the Hamilton-Jacobi-Bellman equation. Additionally, we study existence of minimizers when the reference measure in a Ornstein-Uhlenbeck process.

math.PR↗

arXiv2024

Sharp total variation rates of convergence for fluctuations of linear statistics of $\beta$-ensembles

In this article, we revisit the question of fluctuations of linear statistics of beta ensembles in the single cut and non-critical regime for general potentials $V$ under mild regularity and growth assumptions. Our main objective is to establish sharp quantitative Central Limit Theorems (CLT) for strong distances, such as the total variation distance, which to the best of our knowledge, is new for general potentials, even qualitatively. Namely, setting $\mu_V$ the equilibrium measure, for a test function $\xi \in \mathscr{C}^{14}$, we establish the convergence in total variation of $X_n=\sum_{i=1}^n \xi(\lambda_i)-n\langle \xi,\mu_V\rangle$ to an explicit Gaussian variable at the sharp speed $1/n$. Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in $p-$Wasserstein distances for any $p\ge 1$, with the optimal rate $1/n$, a result which already in dimension one sharpens the speed of convergence established in the recent contribution [26] as well as the required regularity on the test functions. A second objective of this paper, in a more qualitative direction, is to establish the so-called super-convergence of linear statistics, that is to say the convergence of all derivatives of the densities of $X_n$ uniformly on $\mathbb{R}$, provided that $\xi\in\mathscr{C}^\infty(\mathbb{R})$ and is not too degenerated in some sense.

math.PR↗

arXiv2023

Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables

We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in a sequence $X = (X_{i})$ of independent and identically distributed random variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a Gaussian field. This setting encompasses a large class of important and frequently used distributions, such as, among others, Gaussian, Beta, for instance uniform, Gamma distributions, or else any polynomial transform of them. Let us present an emblematic application. Take $X = (X_{i})$ a sequence of independent and identically distributed centered random variables, with unit variance, following such distribution. Consider also $(Q_{n})$ a sequence of quadratic forms, with associated symmetric Hilbert--Schmidt operators $(\mathsf{A}^{(n)})$. Assume that $\operatorname{Tr}[ (\mathsf{A}^{(n)})^{2} ] = 1/2$, $\mathsf{A}^{(n)}_{ii} =0$, and the spectral radius of $\mathsf{A}^{(n)}$ tends to $0$. Then, $(Q_{n}(X))$ converges in a strong sense to the standard Gaussian distribution. Namely, all derivatives of the densities, which are well-defined for $n$ sufficiently large, converge uniformly on $\mathbb{R}$ to the corresponding derivatives of the standard Gaussian density. While classical methods, from Malliavin calculus or $\Gamma$-calculus, generally consist in bounding negative moments of the so-called \emph{carr\'e du champ} operator $\Gamma(Q(X),Q(X))$, we provide a new paradigm through a second-order criterion involving the eigenvalues of a Hessian-type matrix related to $Q(X)$. This Hessian is built by iterating twice a tailor-made gradient, the \emph{sharp operator} $\sharp$, obtained via a Gaussian representation of the carr\'e du champ. We believe that this method, recently developed by the authors in the current paper and in their companion paper [AoP 52 n{\deg}3 (2024)] , is of independent interest and could prove useful in other settings.

math.PR↗

arXiv2023

Superconvergence phenomenon in Wiener chaoses

We establish an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. For every sequence of chaotic random variables, convergence in law to the Gaussian distribution is upgraded to superconvergence: the regularity of the densities increases along the convergence, and all the derivatives converges uniformly. Our findings strengthen known results regarding modes of convergence for normal approximation on Wiener chaoses. Without additional assumptions, convergence in total variation is established by Nourdin & Peccati, and later on amplified to convergence in relative entropy by Nourdin, Peccati & Swan. Our result is then extended to the multivariate setting, and for polynomial mappings of a Gaussian field provided the projection on the Wiener chaos of maximal degree admits a non-degenerate Gaussian limit. While our findings apply to any context involving polynomials of a Gaussian field, we emphasize applications regarding: improved Carbery-Wright estimates near Gaussianity; normal convergence in entropy and in Fisher information; superconvergence for the spectral moments of GOE; moments bounds for the inverse of strongly correlated Wishart-type matrices; superconvergence in the Breuer-Major Theorem. Our proofs leverage Malliavin's historical idea to establish smoothness of the density via the existence of negative moments of the Malliavin gradient, and we develop a new paradigm to study this problem. We relate the existence of negative moments to spectral quantities associated with the Malliavin Hessian. This link relies on an adequate choice of the Malliavin gradient, which provides a novel decoupling procedure of independent interest. Previous attempts to establish convergence beyond entropy have imposed restrictive assumptions ensuring finiteness of negative moments for the Malliavin derivatives. Our analysis renders these assumptions superfluous.

math.PR↗

arXiv2023

Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Let $\varUpsilon$ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure $\pi$. We study the geometry of $\varUpsilon$ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on $\mathscr{P}_{1}(\varUpsilon)$, the space of probability measures over $\varUpsilon$ with finite first moment, and we construct an extended distance $\mathcal{W}$ on $\mathscr{P}_{1}(\varUpsilon)$. The distance $\mathcal{W}$ corresponds, in our setting, to the Benamou--Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with $\mathcal{W}$. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein--Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has a Ricci curvature, in the entropic sense, bounded below by $1$; (c) the distance $\mathcal{W}$ satisfies an HWI inequality.

math.PR↗

arXiv2023

Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous

For an arbitrary dimension $n$, we study: (a) the Polyharmonic Gaussian Field $h_L$ on the discrete torus $\mathbb{T}^n_L = \frac{1}{L} \mathbb{Z}^{n} / \mathbb{Z}^{n}$, that is the random field whose law on $\mathbb{R}^{\mathbb{T}^{n}_{L}}$ given by \begin{equation*} c_n\, e^{-b_n\|(-\Delta_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and $\Delta_{L}$ is the discrete Laplacian; (b) the associated discrete Liouville Quantum Gravity measure associated with it, that is the random measure on $\mathbb{T}^{n}_{L}$ \begin{equation*}\mu_{L}(dz) = \exp \Big( \gamma h_L(z) - \frac{\gamma^{2}}{2} \mathbf{E} h_{L}(z) \Big) dz,\end{equation*} where $\gamma$ is a regularity parameter. As $L\to\infty$, we prove convergence of the fields $h_L$ to the Polyharmonic Gaussian Field $h$ on the continuous torus $\mathbb{T}^n = \mathbb{R}^{n} / \mathbb{Z}^{n}$, as well as convergence of the random measures $\mu_L$ to the LQG measure $\mu$ on $\mathbb{T}^n$, for all $|\gamma| < \sqrt{2n}$.

math.PR↗

arXiv2022

A short proof of the strong three dimensional Gaussian product inequality

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far.

math.PR↗

arXiv2021

Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension

For large classes of even-dimensional Riemannian manifolds $(M,g)$, we construct and analyze conformally invariant random fields. These centered Gaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, are characterized by their covariance kernels $k$ which exhibit a precise logarithmic divergence: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field $h$, we define the quantum Liouville measure, a random measure on $M$, heuristically given as $$ d\mu_g^{h}(x):= e^{\gamma h(x)-\frac{\gamma^2}2k(x,x)}\,d \text{vol}_g(x)$$ and rigorously obtained as almost sure weak limit of the right-hand side with $h$ replaced by suitable regular approximations $h_\ell, \ell\in{\mathbb N}$. In terms on the quantum Liouville measure, we define the Liouville Brownian motion on $M$ and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimensions with an ansatz based on Branson's $Q$-curvature: we give a rigorous meaning to the Polyakov-Liouville measure $$ d\boldsymbol{\nu}^*_g(h) =\frac1{Z^*_g} \exp\Big(- \int \Theta\,Q_g h + m e^{\gamma h} d \text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary compact manifolds.

math.PR↗

arXiv2020

A note on the carr\'e du champ on the Poisson space

The goal of this short note is to establish, in complete generality, the representation for the carr\'e du champ operator associated with the Ornstein-Uhlenbeck semi-group on the Poisson space in terms of the add-one and drop-one operators.

math.PR↗

arXiv2020

Transport inequalities for random point measures

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary ${\sigma}$-finite intensity measure) always satisfies a universal transport-entropy inequality \`a la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [31], originally proved for Poisson random measures with finite mass.

math.PR↗

arXiv2019

Stable limit theorems on the Poisson space

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Sol\'e, Taqqu & Utzet for Gaussian approximations; and by Peccati for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of D\"obler & Peccati. We give an application to stochastic processes.

math.PR↗

arXiv2018

Multiple sets exponential concentration and higher order eigenvalues

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung, Grigory'an and Yau [11].

math.PR↗

arXiv2016

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.

math.DG↗