My main research interest is probability theory.
Office: Campus Irchel Y27J08.

Tightness for the Cover Time of S^{2} (with J. Rosen and O. Zeitouni; preprint)
[ Abstract
arxiv ]
Let C_{ε,S2} denote the cover time of the two dimensional sphere S^{2} by a Wiener sausage of radius ε. We prove that
√C_{ε,S2}−2√2(logε^{1}(1/4)loglog ε^{1}) is tight.
For the critical GaltonWatson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if C_{L} denotes the cover time of the binary tree of depth _{L} by simple walk, then √C_{L}/(2^{L+1})  (√2log2) L + log L/(√2 log2) is tight. The latter improves results of Aldous (1991), Bramson and Zeitouni (2009) and Ding and Zeitouni (2012). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for the twodimensional sphere.
 Barrier estimates for a critical GaltonWatson process and the cover time of the binary tree
(with J. Rosen and O. Zeitouni; preprint)
[ Abstract
arxiv ]
For the critical GaltonWatson process with geometric offspring distributions we provide sharp barrier estimates for barriers which are (small) perturbations of linear barriers. These are useful in analyzing the cover time of finite graphs in the critical regime by random walk, and the Brownian cover times of compact two dimensional manifolds. As an application of the barrier estimates, we prove that if C_{L} denotes the cover time of the binary tree of depth _{L} by simple walk, then √C_{L}/(2^{L+1})  (√2log2) L + log L/(√2 log2) is tight. The latter improves results of Aldous (1991), Bramson and Zeitouni (2009) and Ding and Zeitouni (2012). In a subsequent article we use these barrier estimates to prove tightness of the Brownian cover time for the twodimensional sphere.
 Maximum of the Riemann zeta function on a short interval of the critical line
(with LP. Arguin, P. Bourgade, M. Radziwill and K. Soundararajan; preprint)
[ Abstract
arxiv ]
We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, if t is uniformly distributed in [T,2T], then max t  u ≤ log ζ(1/2 + iu) =(1 + o(1))loglogT
with probability converging to 1 as T → ∞.
 Maximum of the GinzburgLandau fields
(with W. Wu; preprint)
[ Abstract
arxiv ]
We study two dimensional massless field in a box with potential V(∇φ(·)) and zero boundary condition, where V is any symmetric and uniformly convex function. NaddafSpencer and Miller proved the macroscopic averages of this field converge to a continuum Gaussian free field. In this paper we prove the distribution of local marginal φ(x), for any x in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and dimension of high points of this field, thus generalize the results of BolthausenDeuschelGiacomin and Daviaud for the discrete Gaussian free field.
 Maximum of the characteristic polynomial of random unitary matrices
(with LP. Arguin and P. Bourgade; Commun. Math. Phys. Volume 349 (2017), Issue 2, pp 703751.)
[ Abstract
Journal pdf
arxiv ]
It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a N×N random unitary matrix sampled from the Haar measure grows like CN/(logN)^{3/4} for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the range [N^{1−ε},N^{1+ε}], for arbitrarily small ε. The method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other logcorrelated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimensiondependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/fnoise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.
 Maxima of a randomized Riemann Zeta function, and branching random walks
(with LP. Arguin and A. Harper; Ann. Appl. Probab. Volume 27, Number 1 (2017), pp 178215.)
[ Abstract
pdf ]
A recent conjecture of FyodorovHiaryKeating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is exp(loglog T − (3/4) logloglogT + O(1)), for an interval at (large) height T. In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.
 The subleading order of two dimensional cover times. (with N. Kistler; Probab. Theory Relat. Fields, Volume 167 (2017), Issue 1, pp 461552.)
[ Abstract
Journal pdf
arxiv
]
The epsiloncover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance epsilon>0 from any point. Its leading order in the small epsilonregime has been established by Dembo, Peres, Rosen and Zeitouni [Ann. of Math., 160 (2004)]. In this work, the second order correction is identified. The approach relies on a multiscale refinement of the second moment method, and draws on ideas from the study of the extremes of branching Brownian motion.
 Gumbel fluctuations for cover times in the discrete torus. (Probab. Theory Relat. Fields, Volume 157 (2013), Issue 34, pp 635689.)
[ Abstract
Journal pdf
arxiv ]
This work proves that the fluctuations of the cover time of simple random walk in
the discrete torus of dimension at least three with large sidelength are governed by the Gumbel
extreme value distribution. This result was conjectured for example in the book by Aldous & Fill. We also derive some
corollaries which qualitatively describe how covering happens. In addition, we develop a new
and stronger coupling of the model of random interlacements, introduced by Sznitman,
and random walk in the torus. This coupling is used to prove the cover time result and is also
of independent interest.
 Cover times in the discrete cylinder. (Preprint)
[ Abstract
arxiv ]
This article proves that, in terms of local times, the properly rescaled and recentered
cover times of finite subsets of the discrete cylinder by simple random walk
converge in law to the Gumbel distribution, as the cardinality of the set goes to
infinity. As applications we obtain several other results related to covering in the
discrete cylinder. Our method is new and involves random interlacements, which
were introduced in [22]. To enable the proof we develop a new stronger coupling
of simple random walk in the cylinder and random interlacements, which is also of
independent interest.
 Cover levels and random interlacements (Ann. Appl. Probab. Volume 22, Number 2 (2012), pp 522540.)
[ Abstract
pdf ]
This note investigates cover levels of finite sets in the random interlacements model, that is the
least level such that the set is completely contained in the random interlacement at that level. It
proves that as the cardinality of a set goes to infinity, the rescaled and recentered cover level
tends in distribution to the Gumbel distribution with cumulative distribution function exp(exp(z))..