David Belius

Bio: David Belius is an academic researcher from University of Basel. The author has contributed to research in topics: Cover (topology) & Random walk. The author has an hindex of 11, co-authored 29 publications receiving 592 citations. Previous affiliations of David Belius include New York University & ETH Zurich.

Maximum of the Characteristic Polynomial of Random Unitary Matrices

Abstract: 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 \({CN/({\rm log} N)^{3/4}}\) 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 \({\varepsilon}\).

Maximum of the Riemann zeta function on a short interval of the critical line

Abstract: 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→∞ for a set of t∈[T,2T] of measure (1−o(1))T, we have max|t−u|≤1log∣∣ζ(12+iu)∣∣=(1+o(1))loglogT.

Maximum of the characteristic polynomial of random unitary matrices

Abstract: 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.

Maximum of the Riemann zeta function on a short interval of the critical line

Abstract: 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 $$ \max_{|t-u|\leq 1}\log\left|\zeta\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$

Maxima of a randomized Riemann zeta function, and branching random walks

Abstract: 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 Riemann zeta function

Abstract: DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

Fast convolutional sparse coding

Abstract: Sparse coding has become an increasingly popular method in learning and vision for a variety of classification, reconstruction and coding tasks. The canonical approach intrinsically assumes independence between observations during learning. For many natural signals however, sparse coding is applied to sub-elements ( i.e. patches) of the signal, where such an assumption is invalid. Convolutional sparse coding explicitly models local interactions through the convolution operator, however the resulting optimization problem is considerably more complex than traditional sparse coding. In this paper, we draw upon ideas from signal processing and Augmented Lagrange Methods (ALMs) to produce a fast algorithm with globally optimal sub problems and super-linear convergence.

Maximum of the Characteristic Polynomial of Random Unitary Matrices

Abstract: 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 \({CN/({\rm log} N)^{3/4}}\) 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 \({\varepsilon}\).

Branching random walks

Abstract: I Introduction.- II Galton-Watson trees.- III Branching random walks and martingales.- IV The spinal decomposition theorem.- V Applications of the spinal decomposition theorem.- VI Branching random walks with selection.- VII Biased random walks on Galton-Watson trees.- A Sums of i.i.d. random variables.- References.