Schedule
Abstracts
Long courses
Each of the following lecturers delivered four 90 minute lectures:
- Djalil Chafaï - Aspects of random matrices and Coulomb gases
This course is centered around random matrix models and Coulomb gases. After introducing the quarter-circular, the semi-circular,
and the circular laws phenomena, we will study the method of moments, the Stieltjes transform, and the logarithmic potential, which
are efficient tools for the spectral analysis of random matrices. We will then move to the concept of Coulomb gases, and its relation
with large deviations theory and orthogonal polynomials. We will end up with Kesten-McKay laws and the notion of freeness at the
heart of the free probability theory of Voiculescu.
References: http://www.math.polytechnique.fr/xups/xups13-03.pdf and http://arxiv.org/abs/1405.1003
- Roman Kotecký - Gradient models
Gradient Gibbs measures feature in various contexts, notably as models of random surfaces and microscopic models of nonlinear elasticity.
While the aim of the lectures will be to cover recent advances in these areas, we will start with a basic setting
and gradually elucidate technical difficulties involved in studying random gradient fields.
A careful consideration will be given to the case of non-convex interactions. The topics
covered in the lectures will include phase transitions for random surfaces,
variational characterization of nonlinear elasticity obtained as a scaling limit in terms of
large deviations of random gradient fields, as well as an explanation of how to tame
strongly correlated interactions by a multi-scale renormalization group technique.
- Ron Peled - Spatial random permutations (Lecture notes)
Spatial random permutations are non-uniform probability distributions on permutations which
are biased towards the identity with respect to some underlying geometry. One may, for instance, consider random permutations on the points of a finite
metric space $(X,d)$ which are sampled with probability proportional to $\exp\left(-\sum_{x \in X} d(x, \pi(x))\right)$.
A popular model which has been the focus of much
research in recent years is the stirring (or interchange) model. In this model one starts with the identity permutation on the vertices of a (possibly infinite)
graph and lets each edge perform transpositions at the times of an independent Poisson process. The main observable of interest for such models is the
existence of macroscopic (or infinite) cycles.
Spatial random permutation models are related to representations for the quantum Bose gas and macroscopic cycles are associated with the quantum model undergoing Bose-Einstein condensation.
In this mini-course I will describe some of the recent work on spatial random permutations. This includes results by Angel and Schramm (simplified by Berestycki) for the stirring model on a regular tree and on the complete graph, respectively. The well known conjecture for the stirring model on $\mathbb{Z}^d$ is that infinite cycles can appear only in $3$ and higher dimensions and is supported by results by Ueltschi and Betz on certain `annealed' models in $\mathbb{R}^d$. I will also describe results in one dimension with special focus on the Mallows model. Time permitting, I will discuss a second natural observable in one dimension, the longest increasing subsequence, which is related to last passage percolation.
Educational lectures
- Daniel Ueltschi - Random loop models and quantum spin systems
The random loop representations of the quantum Heisenberg models allow to
study these systems using probabilistic methods. They were introduced twenty
years ago by Toth and Aizenman-Nachtergaele. I will explain their derivation
and discuss recent extensions.
If time permits, I will present some rigorous results about a phase transition
with long loops in hypercubes (joint work with R. Kotecky and P. Milos), and about
the decay of certain quantum correlations (joint work with J. Bjornberg).
- Anna Maltsev - Local laws for Wigner random matrices (Presentation)
The spectral measure of Hermitian matrices with centered independent identically distributed entries (Wigner matrices) tends to the semicircle law weakly in the limit of large dimension. This has been proven by Wigner in the 50’s. We ask to what extent this convergence continues to hold on small intervals, i.e. when the interval size tends to 0 with dimension. I will give an overview of the field, mention some recent results, and outline some of the methods.
Short talks by young researchers
- Piotr Dyszewski - Exponential moments of fixed points of the nonhomogeneous smoothing transform
(Presentation)
Consider a (canonical) solution to stochastic fixed point equation $X =^d \sum_k T_k X_k +C$, where $X, X_1, X_2, …$ are iid independent of the random vector $(C, T_1, T_2, …)$. We are interested in necessary and sufficient criteria for the finiteness of exponential moments of $X$ i. e. $\mathbf{E}[ e^{sX}]$. We will provide a formula for the abscissa of convergence of the moment generating function in some special cases, in particular in the case of the random difference equation $R =^d AR+B$. The talk is based on a joint work in progress with Gerold Alsmeyer (University of Muenster).
- Maxime Gagnebin - Decay of correlation in the XY model
We will review the results known about the XY model and show how one can obtain an upper bound on the decay of correlation. We will then see how this bound can be proved for more general models, modifying the range or the type of interaction. Based on joint work with Yvan Velenik.
- Alexey Gladkich - The cycle structure of random Mallows permutations
(Presentation)
The Mallows model is a probability measure on permutations in $S_n$ in which the probability of a permutation is proportional to $q^{inv(\pi)}$, where $inv(\pi)$ denotes the number of inversions in $\pi$ and $q\in (0,1)$ is a parameter of the model. The model is an example of a class of distributions called spatial random permutations in which the distribution is biased to be close to the identity in a certain underlying geometry. We study the cycle structure of a permutation sampled from the Mallows model. Our main result is that the expected length of the cycle containing a given point is of order $\min(1/(1-q)^2, n)$. In contrast, the expected length of a uniformly chosen cycle is of order $\min(1/(1-q),n / \log(n))$. Joint work with Ron Peled.
- Gaultier Lambert - Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
(Presentation)
We will review the central limit theorem for linear statistics of the sine point process from Random Matrix theory. Then, I will explain how the proof can be generalized to a class of determinantal measures in one dimension which interpolate between Poisson and Random Matrix statistics. An example of such a process comes from considering a grand canonical ensemble of free fermions in a quadratic well at positive temperature. For this model, we obtain different Limit theorems for linear statistics depending on the density of the process and the temperature. In particular, in a critical regime, we can observe some non-Gaussian limits. This is a joint work with K. Johansson.
- Thomas Rafferty - Monotonicity and condensation in stochastic particle systems
(Presentation)
Coupling techniques are powerful tools to study hydrodynamic limits of stochastic particle systems. In order for a coupling to exist the process must be monotone, in the sense that the dynamics preserve a partial ordering of the state space for all time. A stochastic process exhibits condensation if above a critical density it phase separates into a homogeneous phase distributed at the maximal Gibbs measure, and a condensate where a diverging number of particles concentrates on a single lattice site. We study stochastic particle systems that conserve particle number and exhibit such condensation transitions due to particle interactions in the limit of diverging mass on a finite lattice. All known examples with product Gibbs measures that exhibit a condensation transition are not monotone, and we prove that this is indeed necessary under the additional assumption of finite first moment of the maximal Gibbs measure. We show that the canonical measures are not monotonically ordered in particle number in case of condensation, which contradicts monotonicity. In the case of infinite first moment, we can construct a model with product stationary measures which is both monotone and condensing. This is joint work with Paul Chleboun and Stefan Grosskinsky.
- Marta Strzelecka - Weak and strong moments of $l_r$-norms of log-concave vectors
(Presentation)
We will discuss the following generalization of the classical Paouris inequality. For $p\geq1$ and $r\geq2$ the $p$-th moment of the $l_r$ norm of a log-concave random vector in $\mathbb{R}^n$ is bounded, up to a constant proportional to $r$, by the sum of its first moment and its $p$-th weak moment. The talk will be based on joint work with Rafał Latała.
- Michał Strzelecki - Modified log-Sobolev inequalities for convex functions
(Presentation)
We will give a sufficient condition for a probability measure on the real line to satisfy a modified logarithmic Sobolev inequality for convex functions. Products of according measures satisfy the two level concentration for convex sets and functions. The talk will be based on joint work with Radosław Adamczak.
- Matija Vidmar - On the information generated by a process up to a stopping time
(Presentation)
There is investigated in detail the notion of the information generated by a (stochastic) process up to a stopping time of its (completed) natural filtration. The two obvious candidates for this body of information the (completed) initial structure of (i.e. the (completed) sigma-field generated by) the stopped process, on the one hand, and the history of the natural filtration of the process up to said stopping time, on the other are shown to be equal under fairly general conditions. This extends existing results in the literature (available for coordinate processes on canonical spaces). Several related findings, results, corollaries and counter-examples are also provided
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