I am Lecturer at the Institute of Mathematics of the University of Zurich.

My main research interest is probability theory.

E-mail: david.belius@cantab.net.

Office: Campus Irchel Y27J08.

- Tightness for the Cover Time of compact two dimensional manifolds
(with J. Rosen and O. Zeitouni;
*submitted*)

[ Abstract arxiv ]Let*C*denote the cover time of the two dimensional sphere_{ε,S2}*S*by a Wiener sausage of radius^{2}*ε*. We prove that*√C*is tight._{ε,S2}−2√2(logε^{-1}-(1/4)loglog ε^{-1}) - Barrier estimates for a critical Galton-Watson process and the cover time of the binary tree
(with J. Rosen and O. Zeitouni; accepted for publication in
*Annales de l’Institut Henri Poincaré*)

[ Abstract arxiv ]For the critical Galton--Watson 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*denotes the cover time of the binary tree of depth_{L}_{L}by simple walk, then*√C*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 two-dimensional sphere._{L}/(2^{L+1}) - (√2log2) L + log L/(√2 log2) - Maximum of the Riemann zeta function on a short interval of the critical line
(with L-P. Arguin, P. Bourgade, M. Radziwill and K. Soundararajan; accepted for publication in
*Communications on Pure and Applied Mathematics*)

[ 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 Ginzburg-Landau 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. Naddaf-Spencer 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 Bolthausen-Deuschel-Giacomin and Daviaud for the discrete Gaussian free field. - Maximum of the characteristic polynomial of random unitary matrices
(with L-P. Arguin and P. Bourgade;
*Communications in Mathematical Physics, Volume 349 (2017), Issue 2, pp 703-751.*)

[ 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 log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols. The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/f-noise 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 L-P. Arguin and A. Harper;
*Annals of Applied Probability, Volume 27, Number 1 (2017), pp 178-215.*)

[ Abstract pdf ]A recent conjecture of Fyodorov--Hiary--Keating 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;
*Probability Theory and Related Fields, Volume 167 (2017), Issue 1, pp 461-552.*)

[ Abstract Journal pdf arxiv ]The epsilon-cover 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 epsilon-regime 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 multi-scale 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 (
*Probability Theory and Related Fields, Volume 157 (2013), Issue 3-4, pp 635-689.*)

[ 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 side-length 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 re-centered 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 (
*Annals of Applied Probability, Volume 22, Number 2 (2012), pp 522-540.*)

[ 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))*..

For the critical Galton--Watson 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 two-dimensional sphere.

- Courant Instructor at the Courant Institute of Mathematical Sciences of New York University.
- CRM-ISM and SNF Postdoctoral Fellow at Centre de Recherches Mathématiques and McGill University. (My hosts were Louigi Addario-Berry and Louis-Pierre Arguin.)
- PhD candidate at ETH Zurich (my adviser was Alain-Sol Sznitman).
- Undergraduate and Master Student (Part III) at Cambridge University.

*Last modified: 26/01/2018*