Rigidity of the Stochastic Airy Operator

Lamarre, Pierre Yves Gaudreau; Ghosal, Promit; Li, Wenxuan; Liao, Yuchen

doi:10.1093/imrn/rnac265

Cited by 2 publications

(2 citation statements)

References 32 publications

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“…This entails investigation of rigidity structures in a wide array point processes that are of interest in probability theory and statistical physics, including the Dyson sine process [22]; the Airy, Bessel, and Gamma processes [3]; and more generally a wide class of determinantal point processes [4,7,9,38,45,52]. Rigidity phenomena have also been investigated in more general settings, such as stationary stochastic processes and random Schrodinger or stochastic Airy operators [5,33,41,42]. Related phenomena, such as appearance of forbidden regions under spatial conditioning [28,29], maximal rigidity [27,39], the relationship between rigidity phenomena and Palm measures [6,23,51], applications to percolation [25,32,37] as well as completeness problems [22], Coulomb and Riesz gases [10,17,18,43,44], random measures and stable matchings [2,40], and directional effects in rigidity and dependency phenomena [1,31] have attracted attention.…”

Section: Singularities Of Conditional Measures and Rigidity Phenomenamentioning

confidence: 99%

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

Gangopadhyay,

Ghosh,

Tan

2023

Comm Pure Appl Math

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Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated random point fields, including those with a highly singular spatial structure. These include processes that exhibit strong spatial rigidity, in particular, a certain one‐parameter family of analytic Gaussian zero point fields, namely the α‐GAFs, that are known to demonstrate a wide range of such spatial behavior. Our framework entails conditions that may be verified via finite particle approximations to the process, a phenomenon that we call an approximate Gibbs property. We show that these enable one to compare the spatial conditional measures in the infinite volume limit with Gibbs‐type densities supported on appropriate singular manifolds, a phenomenon we refer to as a generalized Gibbs property. Our work provides a general mechanism to rigorously understand the limiting behavior of spatial conditioning in strongly correlated point processes with growing system size. We demonstrate the scope and versatility of our approach by showing that a generalized Gibbs property holds with a logarithmic pair potential for the α‐GAFs for any value of α. In this vein, we settle in the affirmative an open question regarding the existence of point processes with any specified level of rigidity. In particular, for the α‐GAF zero process, we establish the level of rigidity to be exactly , a fortiori demonstrating the phenomenon of spatial tolerance subject to the local conservation of moments. For such processes involving complex, many‐body interactions, our results imply that the local behavior of the random points still exhibits 2D Coulomb‐type repulsion in the short range. Our techniques can be leveraged to estimate the relative energies of configurations under local perturbations, with possible implications for dynamics and stochastic geometry on strongly correlated random point fields.

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Section: Singularities Of Conditional Measures and Rigidity Phenomenamentioning

confidence: 99%

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

Gangopadhyay,

Ghosh,

Tan

2023

Comm Pure Appl Math

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“…e βt 3 ), which displays a chaotic nature of the high dimensional multiplicative noise. In a recent work [GGL22], the first author of this paper and his collaborators have introduced the idea of finite time intermittency for the PAM in higher dimension with asymptotically singular noise. We believe that many of our proof techniques can be extended to study the macroscopic fractality of the peaks in those settings.…”

Section: Introductionmentioning

confidence: 99%

Fractal geometry of the PAM in 2D and 3D with white noise potential

Ghosal¹,

Yi²

2023

Preprint

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We study the parabolic Anderson model (PAM)where ξ is spatial white noise on R d with d ∈ {2, 3}. We show that the peaks of the PAM are macroscopically multifractal. More precisely, we prove that the spatial peaks of the PAM have infinitely many distinct values and we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor [BT89, BT92]) of those peaks. As a byproduct, we obtain the exact spatial asymptotics of the solution of the PAM. We also study the spatio-temporal peaks of the PAM and show their macroscopic multifractality. Some of the major tools used in our proof techniques include paracontrolled calculus and tail probabilities of the largest point in the spectrum of the

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Rigidity of the Stochastic Airy Operator

Cited by 2 publications

References 32 publications

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles

Fractal geometry of the PAM in 2D and 3D with white noise potential

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