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Singular distributions, dimension of support, and symmetry of Fourier transform

Gady KozmaAlexander Olevskiĭ — 2013

Annales de l’institut Fourier

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support; (ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l 2 . ...

Entropy of random walk range

Itai BenjaminiGady KozmaAriel YadinAmir Yehudayoff — 2010

Annales de l'I.H.P. Probabilités et statistiques

We study the entropy of the set traced by an -step simple symmetric random walk on ℤ. We show that for ≥3, the entropy is of order . For =2, the entropy is of order /log2. These values are essentially governed by the size of the boundary of the trace.

Supercritical self-avoiding walks are space-filling

Hugo Duminil-CopinGady KozmaAriel Yadin — 2014

Annales de l'I.H.P. Probabilités et statistiques

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory γ between two points on the boundary of a finite subdomain of d is proportional to μ - length ( γ ) . When μ is supercritical (i.e. μ l t ; μ c where μ c is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

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