arXiv subjects
Publications and source records attributed to David Belius.
We reinterpret the Thouless-Anderson-Palmer approach to mean field spin glass models as a variational principle in the spirit of the Gibbs variational principle and the Bragg-Williams approximation. We prove this TAP-Plefka variational principle rigorously in the case of the spherical Sherrington-Kirkpatrick model.
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, as $T \rightarrow \infty$ for a set of $t \in [T, 2T]$ of measure $(1 - o(1)) T$, we have $$ \max_{|t-u|\leq 1}\log\left|ζ\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$
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 $\sqrt{C_L/2^{L+1}} -\sqrt{2\log 2} L+\log L/\sqrt{2\log 2}$ 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.
It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a $N\times N$ random unitary matrix sampled from the Haar measure grows like $CN/(\log N)^{3/4}$ 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 $[N^{1 - \varepsilon},N^{1 + \varepsilon}]$, for arbitrarily small $\varepsilon$. 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.
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 $\exp\{\log \log T -\frac{3}{4}\log \log \log T+O(1)\}$, 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 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.
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.
This note investigates cover levels of finite sets in the random interlacements model introduced in [Ann. of Math. (2) 171 (2010) 2039-2087], 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 $\operatorname {exp}(-\operatorname {exp}(-z))$.
This article proves that, in terms of local times, the 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 by Sznitman in arXiv:0704.2560. 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.