arXiv subjects

Pierre Lazag

Publications and source records attributed to Pierre Lazag.

8 recordsLinked to original sources

arXiv2025

Number variance for homogeneous determinantal processes on hyperbolic spaces

We consider an abstract determinantal point process on a general non--elementary Gromov hyperbolic metric space governed by an orthogonal projection in the case when the space is homogeneous and the point process is invariant under isometries. We give a lower bound of the variance of the number of points inside a ball that is proportional to the volume of the ball. In particular, such point processes are never hyperuniform. Our result applies to the known examples of radial determinantal point processes on Cayley trees and on the standard hyperbolic spaces governed by Bergman projection kernels.

math.PR↗

arXiv2024

Accumulated spectrograms for hyperuniform determinantal point processes

We define the accumulated spectrogram associated to a locally trace class orthogonal projection operator and to a bounded set using the polar decomposition of its restriction on that set and prove a convergence theorem for accumulated spectrograms along an exhaustion in the case when the corresponding determinantal point process is hyperuniform. We prove that a radial determinantal point process on Rd is always hyperuniform along the exhaustion formed by the dilations of a bounded open set, and as a consequence, we obtain that dilations of the corresponding accumulated spectrogram converge to the indicator function of the considered set, establishing thus a universal phenomenon. Our result is a generalisation of a theorem by Abreu-Gröchenig-Romero in [1] concerning time-frequency localization operators.

math.PR↗

arXiv2024

The 2D Toda lattice hierarchy for multiplicative statistics of Schur measures

We prove Fredholm determinants build out from generalizations of Schur measures, or equivalently, arbitrary multiplicative statistics of the original Schur measures are tau-functions of the 2D Toda lattice hierarchy. Our result apply to finite temperature Schur measures, and extends both the result of Okounkov in \cite{okounkovschurmeasures} and of Cafasso-Ruzza in \cite{cafassoruzza} concerning the finite-temperature Plancherel measure. Our proof lies on the semi-infinite wedge formalism and the Boson-Fermion correspondance.

math-ph↗

arXiv2024

Explicit description of Christoffel deformations and Palm measures of the Plancherel measure, the $z$-measures and the Gamma process

Christoffel deformation of a measure on the real line consists of multipying this measure by a squared polynomial having its roots in $\R$. We introduce Christoffel deformations of discrete orthogonal polynomial ensembles by considering the Christoffel deformations of the underlying measure, and prove that this construction extends to more general point processes describing distribution on partitions: the poissonized Plancherel measure and the $z$-measures. These deformations contain the theory of Palm measures, and for example, explicit formulas for the Palm measures of the poissonized Plancherel measure provide a description of the TASEP with initial wedge condition with frozen particles. We also obtain new formulas for Palm measures of the $z$-measures. The extension to the Plancherel measure is obtained via a limit transition from the Charlier ensemble, while the extension to the $z$-measures follows from an analytic continuation argument. A limit procedure starting from the non-degenerate $z$-measures leads to a deformation of the Gamma process introduced by Borodin and Olshanski.

math.PR↗

arXiv2022

A law of large numbers for local patterns in Schur measures and a Schur process

The aim of this note is to prove a law of large numbers for local patterns in discrete point processes. We investigate two different situations: a class of point processes on the one dimensional lattice including certain Schur measures, and a model of random plane partitions, introduced by Okounkov and Reshetikhin. The results state in both cases that the linear statistic of a function, weighted by the appearance of a fixed pattern in the random configuration and conveniently normalized, converges to the deterministic integral of that function weighted by the expectation with respect to the limit process of the appearance of the pattern.

math.PR↗

arXiv2021

A determinantal point process governed by an integrable projection kernel is Giambelli compatible

The first main result of this note, Theorem 1.2, establishes the determinantal identities (7) and (8) for the expectation, under a determinantal point process governed by an integrable projection kernel, of scaling limits of characteristic polynomials sampled at several points. The determinantal identities (7) and (8) can be seen as the scaling limit of the identity of Fyodorov and Strahov for the averages of ratios of products of the values of the characteristic polynomial of a Gaussian unitary matrix. Borodin, Olshanski and Strahov derived the determinantal identity of Fyodorov and Strahov from the stability of the Giambelli formula under averaging. In Theorem 1.4 the stability of the Giambelli formula under averaging is established for determinantal point process with integrable projection kernels. The proof of Theorems 1.2 and 1.4 relies on the characterization of conditional measures of our point processes as orthogonal polynomial ensembles.

math.PR↗

arXiv2020

A law of large numbers for local patterns in plane partitions

In this note, we prove a law of large numbers for local patterns in plane partitions with geometric weight, a model introcued by Okounkov and Reshetikhin. Its proof is based on the determinantal structure of the corresponding point process, wich allows to control the decay of the correlations in a convenient manner.

math.PR↗

arXiv2017

The Hyperbolic-type Point Process

In this paper, we introduce a two-parameters determinantal point process in the Poincaré disc and compute the asymptotics of the variance of its number of particles inside a disc centered at the origin and of radius $r$ as $r$ tends to $1$. Our computations rely on simple geometrical arguments whose analogues in the Euclidean setting provide a shorter proof of Shirai's result for the Ginibre-type point process. In the special instance corresponding to the weighted Bergman kernel, we mimic the computations of Peres and Virag in order to describe the distribution of the number of particles inside the disc.

math.PR↗