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Author: Avena, Luca ×

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    On some random forests with determinantal roots
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Avena, Luca; Gaudillière, Alexandre
    Consider a finite weighted oriented graph. We study a probability measure on the set of spanning rooted oriented forests on the graph. We prove that the set of roots sampled from this measure is a determinantal process, characterized by a possibly non-symmetric kernel with complex eigenvalues. We then derive several results relating this measure to the Markov process associated with the starting graph, to the spectrum of its generator and to hitting times of subsets of the graph. In particular, the mean hitting time of the set of roots turns out to be independent of the starting point, conditioning or not to a given number of roots. Wilson's algorithm provides a way to sample this measure and, in absence of complex eigenvalues of the generator, we explain how to get samples with a number of roots approximating a prescribed integer. We also exploit the properties of this measure to give some probabilistic insight into the proof of an algebraic result due to Micchelli and Willoughby [13]. Further, we present two different related coalescence and fragmentation processes.
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    The parabolic Anderson model on the hypercube
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Avena, Luca; Gün, Onur; Hesse, Marion
    We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and the k-th largest potential. One of our main motivations in this article is to investigate the mutationselection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.