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# Fribergh, Alexander

Associate Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-6709

### Affiliations

• Membre Centre de recherches mathématiques
• Membre CRM  Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Extremes of log-correlated random fields and the Riemann zeta function, and some asymptotic results for various estimators in statistics Theses and supervised dissertations / 2019-05
Ouimet, Frédéric
Abstract
In this thesis, we study the extreme values of certain log-correlated random fields that are Gaussian (the scale-inhomogeneous Gaussian free field and the time-inhomogeneous branching random walk) or approximatively Gaussian (the log-modulus of the Riemann zeta function on the critical line and a randomized toy model of it), as well as asymptotic properties of various estimators in statistics. Apart from the introduction and conclusion, the thesis is divided in three parts, each containing three articles. The first part contains three articles on log-correlated Gaussian fields. The first article shows the first order convergence of the maximum and the number of high points for the scale-inhomogeneous Gaussian free field on its full domain. The second article uses the results from the first article to show that the limiting law of the Gibbs measure is a Ruelle probability cascade with a certain number of effective scales (a tree of Poisson-Dirichlet processes). The third article shows the tightness of the recentered maximum for the time-inhomogeneous branching random walk. The second part contains three articles on the Riemann zeta function. The first article shows that, at low temperature, the limiting law of the Gibbs measure for a randomized toy model of the log-modulus of zeta on the critical line is a Poisson-Dirichlet process. The second article deals with the open problem of the tightness of the recentered maximum for this toy model on an interval of length $O(1)$. We simplify the problem by showing that the continuous maximum is at the order of constant away from a discrete maximum over $O(\log T \sqrt{\log \log T})$ points. The third article shows the first order of convergence of the maximum and the free energy for the log-modulus of the Riemann zeta function on short intervals of length $O(\log^{\theta} T)$, $\theta > -1$, on the critical line. The third part contains three articles treating various topics in asymptotic statistics. The first article shows the complete monotonicity of multinomial probabilities and opens the door to the study of the asymptotic properties of Bernstein estimators on the simplex. The second article shows a uniform law of large numbers for sums containing terms that blow up''. The third article finds the limiting law of a modified score statistic when we test a given member of the exponential power distribution family against the family of asymmetric power distributions. The thesis contains nine articles of which seven are already published in peer-reviewed journals. All the information is gathered on my personal website : https://sites.google.com/site/fouimet26/research.

Étude de la marche aléatoire biaisée en milieu aléatoire Theses and supervised dissertations / 2017-11
Laliberté, Nicolas
Abstract
The main goal of this Master thesis is to provide an introduction to the random walks in random environments (RWRE). This is a vast domain, our interest will be on models where the walk is directionnally transient in order to study asymptotical behavior. We will present severals models and associated mains results. We will see some basic techniques used in this field. One of the tools that has made the most progress is the renewal structure which allows us to answer questions about the speed of the biased random walk. This is the subject of this thesis, we deal with the following question : Is the speed of the biaised random walk on a Galton-Watson tree monotonous with respect to the environnement ? We answer in the affirmative for a high bias.

Vieillissement pour la marche aléatoire biaisée sur des conductances aléatoires dans l'hyper-grille à d dimensions Theses and supervised dissertations / 2017-10
Davignon, Thomas
Abstract
We examine the property of ageing exhibited by several models of random walk on random environments that display trapping behavior. In particular we present a proof of such a property for the biased random walk on random conductances in the d-dimensional hyper-cubic lattice, when the conductances are independant and heavy-tailed (i.e. no first moment).

Théorème Central Limite pour les marches aléatoires biaisées sur les arbres de Galton-Watson avec feuilles Theses and supervised dissertations / 2016-09
Rakotobe, Joss
Abstract
This Master thesis is part of a larger project of linking the behaviours of a certain type of random walks in random environments (RWRE) with those of a toy model called the Bouchaud’s trap model. The domain of RWRE is very wide but our interest will be on a particular kind of models which are reversible and directionally transient. More specifically, we will see why those models have similar behaviours and what kind of results we could expect once we have reviewed the Bouchaud’s trap model. We will also present some basic technic used in this field, such as regeneration times. As a contribution, we will demonstrate a central limit theorem for the β-biased random walk on a Galton-Watson tree.

### Selected publications Expand allCollapse all

#### Scaling limit for the ant in a high-dimensional labyrinth

Gérard Ben Arous, Manuel Cabezas, Alexander Fribergh, Scaling limit for the ant in a high-dimensional labyrinth 72(1) , (2019), , Communications in Pure and Applied Mathematics

#### Scaling limits for sub-ballistic biased random walks in random conductances

Alexander Fribergh et Daniel Kious, Scaling limits for sub-ballistic biased random walks in random conductances 46(2), 605-686 (2018), , Annals of Probability

#### Scaling limit for the ant in a simple labyrinth

Gérard Ben Arous, Manuel Cabezas et Alexander Fribergh, Scaling limit for the ant in a simple labyrinth , 94 (2018), , Annals of probability

#### Biased random walk on the interlacement set

Alexander Fribergh et Serguei Popov, Biased random walk on the interlacement set 54, (2018), , Annales de l'institut Henry Poincaré

#### Local trapping for elliptic random walks in random environments in Zd

Alexander Fribergh and Daniel Kious, Local trapping for elliptic random walks in random environments in Zd 165, 795--834 (2016), , Probability Theory and Related Fields

#### Scaling limits for random walks on random critical trees

Ben Arous, Cabezas, Fribergh, Scaling limits for random walks on random critical trees , (2016), , Annales de l'Institut Poincaré

#### Biased random walks on random graphs

Gérard Ben Arous and Alexander Fribergh, Biased random walks on random graphs , (2015), , AMS volume Saint Petersburg summer School on Probability and Statistical Mechanics edited by S. Smirnov and V. Sidoravicius

#### Lyons-Pemantle-Peres monotonicity problem for high biases

Ben Arous, Gérard, Fribergh, Alexander et Sidoravicius, Vladas, Lyons-Pemantle-Peres monotonicity problem for high biases 67, 519--530 (2014), , Comm. Pure Appl. Math.

#### Phase transition for the speed of the biased random walk on the supercritical percolation cluster

Fribergh, Alexander et Hammond, Alan, Phase transition for the speed of the biased random walk on the supercritical percolation cluster 67, 173--245 (2014), , Comm. Pure Appl. Math.

#### Biased random walk in positive random conductances on $\Bbb Z^d$

Fribergh, Alexander, Biased random walk in positive random conductances on $\Bbb Z^d$ 41, 3910--3972 (2013), , Ann. Probab.

#### Biased random walk on critical Galton-Watson trees conditioned to survive

Croydon, D. A., Fribergh, A. et Kumagai, T., Biased random walk on critical Galton-Watson trees conditioned to survive 157, 453--507 (2013), , Probab. Theory Related Fields

#### Biased random walks on Galton-Watson trees with leaves

Ben Arous, Gérard, Fribergh, Alexander, Gantert, Nina et Hammond, Alan, Biased random walks on Galton-Watson trees with leaves 40, 280--338 (2012), , Ann. Probab.

#### The speed of a biased random walk on a percolation cluster at high density

Fribergh, Alexander, The speed of a biased random walk on a percolation cluster at high density 38, 1717--1782 (2010), , Ann. Probab.

#### On slowdown and speedup of transient random walks in random environment

Fribergh, Alexander, Gantert, Nina et Popov, Serguei, On slowdown and speedup of transient random walks in random environment 147, 43--88 (2010), , Probab. Theory Related Fields