Geometry of the Gibbs measure for the discrete 2D
Gaussian free field with scale-dependent variance

Ouimet, Frédéric

doi:10.30757/alea.v14-38

Cited by 6 publications

(4 citation statements)

References 72 publications

(105 reference statements)

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“…More precisely, it is proved that a measure similar to ν n,β /ν n,β (U ) converges in law toward a Ruelle probability cascade. Ouimet [Oui17] recently extended this family of results to Gaussian fields with scale-dependent variance. Theorems 4.1 and 4.3 represent extensions of these results to branching random walks with non-Gaussian increments.…”

Section: The Supercritical Gibbs Measurementioning

confidence: 92%

Genealogy of the extremal process of the branching random walk

Mallein¹

2018

ALEA

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The extremal process of a branching random walk is the point measure recording the position of particles alive at time n, shifted around the expected position of the minimal position. Madaule [Mad17] proved that this point measure converges, as n → ∞, toward a randomly shifted, decorated Poisson point process. In this article, we study the joint convergence of the extremal process together with its genealogical informations. This result is then used to describe the law of the decoration in the limiting process, as well as to study the supercritical Gibbs measures of the branching random walk.

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Section: The Supercritical Gibbs Measurementioning

confidence: 92%

Genealogy of the extremal process of the branching random walk

Mallein¹

2018

ALEA

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“…The idea for the statement originates from [8], and the idea behind the proof generalizes the special-case application in [5]. See [6,20] for an application in the context of the Gaussian free field.…”

Section: )mentioning

confidence: 99%

Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Ouimet¹

2018

Electron. Commun. Probab.

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In [4], the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.

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“…For other recent and relevant results on branching processes in time-inhomogeneous environments, the reader is referred to Hartung (2014, 2015), Kurkova (2004a, 2004b), Chen (2018), Fang and Zeitouni (2012b), Maillard and Zeitouni (2016), Mallein (2015a), Mallein and Piotr (2015), Ouimet (2017).…”

Section: Related Workmentioning

confidence: 99%

Maxima of branching random walks with piecewise constant variance

Ouimet¹

2018

Braz. J. Probab. Stat.

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This article extends the results of Fang and Zeitouni [Electron. J. Probab. 17 (2012a) 18] on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0, 1] under a slight restriction. We find the asymptotics of the maximum up to an O P (1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein [Electron. J. Probab. 20 (2015b) 40] when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang and Zeitouni [Electron. J. Probab. 17 (2012a) 18] instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.

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Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

Cited by 6 publications

References 72 publications

Genealogy of the extremal process of the branching random walk

Genealogy of the extremal process of the branching random walk

Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Maxima of branching random walks with piecewise constant variance

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