Schedule

Location: Mathematikzentrum, Endenicher Allee 60, Lipschitz Lecture Hall

Location: Mathematikzentrum, Endenicher Allee 60, Lipschitz Lecture Hall

Location: Max Planck Institute, Vivatsgasse 7, MPIM Lecture Hall

Location: Mathematikzentrum, Endenicher Allee 60, Lipschitz Lecture Hall

A connection between branching structures and characteristic polynomials of random matrices emerged in the past few years. We will illustrate this for two models of random matrices, corresponding to dimension 1 and 2 spectra: the Circular Unitary Ensemble and Ginibre random matrices. The discussed topics will include the second moment method, extrema, the Gaussian free field and Gaussian multiplicative chaos.

Branching Brownian motion (BBM) is a classical process in probability theory, describing a population of particles performing independent Brownian motion and branching according to a Galton-Watson process. It also belongs the class of so called log-correlated random fields. We will focus on the behaviour of the extremal particles of BBM. First, we will understand how the correlations in the model effect the order of the maximum. Then, I will explain why the extremal process of BBM converges to a random cluster process. Building on these known results, we will move on to recent results on the extreme level sets of BBM. We find the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. I will explain how truncated moments help to reduce questions on the size / shape of extreme level sets to random walk-like estimates.

I will review some aspects of the theory of the Riemann zeta function, especially those aspects relating to the value distribution of the zeta function on the critical line and the connections with random matrix theory. I will then review conjectural connections between the extreme value statistics of the zeta function and those of log-correlated Gaussian fields, recent progress towards developing a rigorous understanding of these, and open problems.

In this talk, we will present a proof of the freezing transition for the Riemann zeta function as conjectured by Fyodorov, Hiary & Keating. The connection with logcorrelated fields will be emphasized. The problem is related to understanding moments of zeta on a typical short interval. The proof relies on techniques developed to understand the leading order of the maximum of zeta. If time permits, we will discuss the “one-step replica symmetry breaking behaviour” (1-RSB) which can be proved for a simplified model of zeta.

This talk is based on a joint work with J. Najnudel, currently in progress, where we wish to relate two questions or rather two integrable models.
On the one hand, in 1985, J.P Kahane introduced a random measure called the Gaussian multiplicative chaos. It is morally the measure whose Radon-Nikodym derivative with respect to Lebesgue is the exponential of a Gaussian Free Field. A nice martingale argument allows to define the object, but certain of its properties remain inaccessible. This object seems to be at the heart of recent works (Liouville quantum gravity by Rhodes, Vargas, Duplantier, Sheffield...). We are interested in the case of the circle, which is a very integrable geometry.
On the other hand, it is known since Verblunsky (1930s) that a measure on the circle is entirely determined by certains coefficients called the reflection or Verblunsky coefficients. In simple terms, these appear in the recurrence of orthogonal polynomials.
I shall present a conjecture which qualifies precisely the distribution of Verblunsky coefficients for the Gaussian Multiplicative Chaos, and the partial results in that direction. Our results come from analyzing a random matrix model, the circular beta ensemble, closely tied to the trigonometric Calogero-Moser system.

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher-KPP equation with random branching and its linearized version the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher-KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in O(ln t). If time admits we will address some work in progress as well as open questions the answer to which is currently eluding us.

I will discuss some new results concerning extreme and large values of the 2D discrete Gaussian free field and related processes. These include finer structural properties of its extremal landscape, scaling limits for its high (but not extreme) level sets and the asymptotic growth of the infinite volume pinned DGFF. Based on joint work (some in progress) with M. Biskup, A. Cortines, L. Hartung and D. Yeo.

An infinitely ramified point measure is a random point measure that can be written as the terminal value of a branching random walk of any length. This is the equivalent, in branching processes theory, to the notion of infinitely divisible random variables for real-valued random variables. In this talk, we show a connexion between infinitely ramified point measures and branching Lévy processes, a continuous-time particle system on the real line, in which particles move according to independent Lévy processes, and give birth to children in a Poisson fashion.

We show that under the Riemann hypothesis, the ratio by log log T of the maximum of the real or the imaginary part of the Riemann zeta function on a segment of fixed length of the critical line, centered at a uniform point of the segment [1/2, 1/2 + iT], tends to 1 in probability when T goes to infinity. This proves a partial version of a conjecture by Fyodorov, Hiary and Keating.

In the 1990s, S. Kerov described the limiting fluctuations of several natural random interlacing pairs, particularly, the eigenvalues of a random matrix and a principal submatrix, and the outer and inner corners of a random Young diagram. We shall discuss these examples along with a new one: the eigenvalues of a random matrix and the critical points of its characteristic polynomial.

Based on the rigorous path integral formulation of Liouville conformal field theory introduced by David-Kupiainen-Rhodes-Vargas, we compute the first order asymptotic of the four-point correlation function on the sphere as two insertions get close together, which is expected to describe the density of vertices around the root of large random planar maps. Moreover, our results are consistent with predictions from the framework of theoretical physics known as the conformal bootstrap, featuring the DOZZ formula and logarithmic corrections in the distance between the insertions.

The discrete Gaussian Free Field on the lattice is a logarithmically correlated Gaussian random field. In recent years, there has been a lot of progress in the research of its extremes which is still going on. We are studying a generalized version proposed by Louis-Pierre Arguin and Olivier Zindy, where one considers a logarithmically correlated Gaussian random field with a scale-dependent covariance structure. This may be considered as an analogue model to the time-inhomogeneous branching random walk. We are going to present this model and give first results on the maximum value of this field.

Motivated by the Lee and Yang theory of phase transitions, quantum physics formalism, relationships with Riemann's zeta function, we consider the complexvalued random energy model (REM), generalized REM (GREM), and the branching Brownian motion (BBM) energy model. The latter is a log-correlated random field. We identify the respective phase diagrams, study the fluctuations of the partition function and compute the log-partition functions. We allow for arbitrary correlations between real and imaginary parts of the energies. It turns out that the complex BBM energy model lies exactly at the borderline of the complex REM universality class. Based on joint works with Zakhar Kabluchko and Lisa Hartung.

In recent years the strong links between the geometry of smooth planar Gaussian fields and percolation have become increasingly apparent, and it is now believed that the connectivity of the level sets of a wide class of smooth, stationary planar Gaussian fields exhibits a sharp phase transition that is analogous to the phase transition in, for instance, Bernoulli percolation. In recent work we prove this conjecture under the assumptions that the field is (i) symmetric, (ii) positively correlated, and (iii) has an integrable covariance kernel. Key to our techniques are (i) the white-noise representation of Gaussian fields, and (ii) the randomised algorithm approach to noise sensitivity. Joint work with Dmitry Beliaev, Hugo Vanneuville and Alejandro Rivera.

Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will explain how to prove rigorously this formula by using the techniques of conformal field theory. The key observation is that the moments of the total mass of GMC on the circle are equal to one-point correlation functions of Liouville theory in the unit disk. The same techniques also allow to derive a similar result on the unit interval [0,1] (in collaboration with Tunan Zhu). Finally we will briefly discuss applications to random matrix theory and to the asymptotics of the maximum of the GFF.

We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two finite moments, in which all particles have a drift towards the origin and are immediately killed if they reach it. It is well-known that if and only if the branching rate is sufficiently large, the population survives forever with a positive probability. We show that throughout this super-critical regime, the number of particles inside any given set normalized by the mean population size converges to an explicit limit, almost surely and in L1. As a consequence, we get that, almost surely on the event of survival of the branching process, the empirical distribution of particles converges weakly to the (minimal) quasi-stationary distribution associated with the Markovian motion driving the particles. This proves a result of Kesten from 1978, for which no proof was available until now. Joint work with Oren Louidor.

Joseph Najnudel will play the Beethoven pieces

Moonlight sonata in C sharp minor, op. 27 no. 2.

Appassionnata sonata in F minor, op. 57.

and some of his own compositions.