# Probability Seminar

##### Organisers

Leonardo T. Rolla and Vedran Sohinger in collaboration with Sasha Sodin (Queen Mary), Jess Jay and Benjamin Lees (Bristol).

##### Seminars in Term 1

Oct 13 - Alexander Povolotsky, Joint Institute for Nuclear Research, Dubna, Russia.

Title: Generalized TASEP between KPZ and jamming regimes

Abstract: Totally Asymmetric Simple Exclusion Process (TASEP) with generalized update is an integrable stochastic model of interacting particles, which differs from the standard TASEP by the presence of an additional interaction controlling the degree of particle clustering. As the strength of the interaction varies, the system suffers the transition from the regime in which the fluctuations of the particle flow are described by standard random processes associated with the Kardar-Parisi-Zhang (KPZ) universality class, to the jamming regime, in which all particles stick to one cluster and move synchronously as the simple random walk. I will focus on the limiting laws of fluctuations of distances travelled by tagged particles both in the KPZ and in the transitional regime for two types of initial conditions. In particular new transitional processes interpolating between the known limiting cases will be discussed.

Oct 20 - Eveliina Peltola, University of Bonn & Aalto University

Title: Large deviations of SLEs, real rational functions, and zeta-regularized determinants of Laplacians

Abstract: When studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, we recently introduced a ''Loewner potential'' that describes the rate function for the LDP. This object turned out to have several intrinsic, and perhaps surprising, connections to various fields. For instance, it has a simple expression in terms of zeta regularized determinants of Laplace-Beltrami operators. On the other hand, minima of the Loewner potential solve a nonlinear first order PDE that arises in a semiclassical limit of certain correlation functions in conformal field theory,
arguably also related to isomonodromic systems. Finally, and perhaps most interestingly, the Loewner potential minimizers classify rational functions with real critical points, thereby providing a novel proof for
a version of the now well-known Shapiro-Shapiro conjecture in real enumerative geometry. This talk is based on joint work with Yilin Wang (MIT).

Oct 27 - Matan Harel, Northeastern University

Title: Quantitative estimates on the effect of disorder on low-dimensional lattice models

Abstract: In their seminal work, Imry and Ma predicted that the addition of an arbitrarily small random external field to a low-dimensional statistical physics model causes the usual first-order phase transition to be `rounded-off.' This phenomenon was proven rigorously by Aizenman and Wehr in 1989 for a vastly general class of spin systems and random perturbations. Recently, the effect was quantified for the random-field Ising model, proving that it exhibits exponential decay of correlations at all temperatures. Unfortunately, the analysis relies on the monotonicity (FKG) properties which are not present in many other classical models of interest. This talk will present quantitative versions of the Aizenman-Wehr theorems for general spin systems with random disorder, including Potts, spin O(n), spin glasses. This is joint work with Paul Dario and Ron Peled.

Nov 03 - Ariel Yadin, Ben-Gurion University

Title: Realizations of random walk entropy

Abstract: Random walk entropy is a numerical measure of the behaviour of the random walk at infinity. Out of all random walks on groups generated by d elements, the free group has the maximal entropy. One may ask naturally which intermediate values between 0 and the full entropy of the free group can be realized as entropies of random walks on groups. This question is still open. We analyze a related question, which is a "stochastic" version of the above open question. Here we are able to provide a full answer for quotients of the free group, and even a bit further than that. Generalizing results of Bowen (Inventiones 2014), we show that all possible "IRS entropy" values can be realized on the free group. These notions will be precisely explained during the talk. Based on joint works with Yair Hartman and Liran Ron-George.

Nov 10 - Slim Kammoun, Université de Toulouse III (France).

Title: Longest Common Subsequence of Random Permutations

Abstract: Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence (LCS) of two i.i.d random permutations of size $n$ is greater than $\sqrt{n}$.

This problem is related to the Ulam-Hammersley problem; Ulam conjectured that the expectation of the length of the longest increasing subsection (LIS) for a uniform permutation behaves like $c\sqrt{n}$. The conjecture was solved in 1977, but few results are known for non-uniform permutations. The LIS and LCS are closely related, and solving the conjecture of Bukh and Zhou is equivalent to minimize the expected value of LIS for random permutations that can be written as $\rho_n\circ\sigma_n^{-1}$ where $\sigma_n$ and $\rho_n$ are i.i.d. random permutations.

We recall the classical results for the uniform case as well as partial answers for the conjugation invariant case.

Nov 17 - Fabio Toninelli, Technical University of Vienna.

Title: Diffusion in the curl of the 2-dimensional Gaussian Free Field

Abstract: I will discuss the large time behaviour of a Brownian diffusion in two dimensions, whose drift is divergence−free, ergodic and given by the curl of the 2−dimensional Gaussian Free Field. Together with G. Cannizzaro and L. Haundschmid, we prove the conjecture by B. Toth and B. Valko that the mean square displacement is of order t√log t. The same type of superdiffusive behaviour has been predicted to occur for a wide variety of (self)−interacting diffusions in dimension d = 2: the diffusion of a tracer particle in a fluid, self−repelling polymers and random walks, Brownian particles in divergence−free random environments, and, more recently, the 2−dimensional critical Anisotropic KPZ equation. To the best of our authors’ knowledge, ours is the first instance in which √log t superdiffusion is rigorously established in this universality class.

Nov 24 - Sarah Penington, University of Bath.

Title: Genealogy of the N-particle branching random walk with polynomial tails

Abstract: The N-particle branching random walk is a discrete time branching particle system with selection consisting of N particles located on the real line. At every time step, each particle is replaced by two offspring, and each offspring particle makes a jump from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. I will discuss recent results and open conjectures about the long-term behaviour of this particle system when N, the number of particles, is large. In the case where the jump distribution has regularly varying tails, building on earlier work of J. Bérard and P. Maillard, we prove that at a typical large time the genealogy is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale. Based on joint work with Matt Roberts and Zsófia Talyigás.

Dec 01 - Ofer Zeitouni, Weizmann Institute of Science & NYU

Title: High moments of partition function for 2D polymers in the weak disorder regime.

Abstract: Consider ${W}_{N}\left(\beta ,x\right)={\mathrm{E}}_{x}\left[{e}^{\sum _{n=1}^{N}\beta \omega \left(n,{S}_{n}\right)-N{\beta }^{2}/2}\right]$ for $x\in {\mathbb{Z}}^{d}$ with $d=2$ , where the $\omega \left(i,x\right)$ are assumed to be iid centered Gaussians and ${S}_{n}$ is simple random walk. This is a directed polymer partition function. Take ${\beta }_{N}=\frac{\stackrel{^}{\beta }}{\sqrt{{R}_{N}}}$ with ${R}_{N}={\mathrm{E}}_{0}^{\otimes 2}\left[\sum _{n=1}^{N}{\mathbf{1}}_{{S}_{n}^{1}={S}_{n}^{2}}\right]\sim \left(\mathrm{log}N\right)/\pi .$ Write ${W}_{N}={W}_{N}\left({\beta }_{N},0\right)$ .

Caravena, Sun and Zygouras showed that for all $\stackrel{^}{\beta }<1$ , $\mathrm{log}{W}_{N}\stackrel{\left(d\right)}{⟶}\mathcal{N}\left(-\frac{{\lambda }^{2}}{2},{\lambda }^{2}\right)$ , with ${\lambda }^{2}\left(\stackrel{^}{\beta }\right)=-\mathrm{log}\left(1-{\stackrel{^}{\beta }}^{2}\right)$ . They also showed that $R N ( log ⁡ W N ( β N , x N ) − E log ⁡ W N ( β N , x N ) ) ⟶ ( d ) β ^ 2 1 − β ^ 2 G ( x ) ,$ with $G\left(x\right)$ a log-correlated Gaussian field on ${\mathbb{R}}^{2}$ . With an eye toward constructing multiplicative chaoses based on $\sqrt{{R}_{N}}\mathrm{log}{W}_{N}\left({\beta }_{N},x\sqrt{N}\right)$ , it is natural to consider the $q$ -moments ${r}_{q,N}=\mathbb{E}{W}_{N}^{q}$ , with $q\sim \sqrt{\mathrm{log}N}$ .

Lygkonis and Zygouras have recently shown that for $q$ finite independent of $N$ , ${r}_{q,N}$ coincide with the exponential moments of Gaussians. In this talk, I will describe joint work with Clement Cosco, where we prove the following.

There exists ${\stackrel{^}{\beta }}_{0}\in \left(0,1\right)$ and $\alpha \in \left(0,1\right)$ such that for all $\stackrel{^}{\beta }<{\stackrel{^}{\beta }}_{0}$ and ${q}^{2}{\lambda }^{2}\le \alpha \mathrm{log}N$ , one has: $r q , N ≤ e ( q 2 ) λ 2 1 1 − r ( 1 + | ε N | ) ,$ with $r=12\left(\genfrac{}{}{0}{}{q}{2}\right)\frac{1}{\mathrm{log}N}\frac{{\stackrel{^}{\beta }}^{2}}{1-{\stackrel{^}{\beta }}^{2}}<1$ and ${\epsilon }_{N}=\epsilon \left(N,\stackrel{^}{\beta }\right)\to 0$ as $N\to \mathrm{\infty }$ . In particular, for all $\stackrel{^}{\beta }<{\stackrel{^}{\beta }}_{0}$ , uniformly for ${q}^{2}=o\left(\mathrm{log}N\right)$ , $r q , N ≤ e ( q 2 ) λ 2 ( 1 + o ( 1 ) ) .$

With the same method, we also prove that the estimate [above] holds for all $\stackrel{^}{\beta }<1$ at the cost of choosing ${q}^{2}=o\left(\mathrm{log}N/\mathrm{log}\mathrm{log}N\right)$ , giving (together with the Gaussian convergence of CSZ) an independent proof of the Lygkonis-Zygouras theorem. I will discuss some background, the ideas of the proof, potential extensions, and open questions.

Dec 08 - Cyril Labbé, Paris Dauphine University.

Title: Mixing points on an interval

Abstract: Consider N points on the unit interval. Resample each point at rateone uniformly in between its two nearest neighbours. The question is: how much time is needed to reach equilibrium (the uniform measureonthe simplex) starting from the worst initial condition ? I will present results in collaboration with Pietro Caputoand Hubert Lacoin where we identified the asymptotic of themixingtimesand showed a cutoff phenomenon for the distance to equilibrium.

Jan 12 - Zied Ammari, Université Rennes 1.

Title: On well-posedness for the Gross-Pitaevskii and Hartree Hierarchy Equations

Abstract: Gross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs related to mean field theory of Bose gases. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these hierarchies have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this talk I will introduce a new approach based on a duality between hierarchies and Liouville equations. Several new results are obtained as an outcome of this approach.

Jan 19 - Shahar Mendelson, University of Warwick.

Title: Approximating Lₚ balls via sampling

Abstract: Let X be a centred random vector in Rⁿ. The Lₚ norms that X endows on Rⁿ are defined by ‖v‖_L__ₚ= (E|<X,v>|ᵖ)¹/ᵖ. The goal is to approximate those Lₚ norms, and the given data consists of N independent sample points X₁,...,X_N distributed as X. More accurately, one would like to construct data−dependent functionals ϕₚ,ε which satisfy with (very) high probability, that for every v in Rⁿ, (1−ε) ϕₚ,ε ≤ E|<X,v>|ᵖ ≤ (1+ε) ϕₚ,ε. I will show that the functionals \frac{1}{N}∑ⱼ∈J |<Xⱼ,v>|ᵖ are a good choice, where the set of indices J is obtained from {1,...,N} by removing the cε²N largest values of |<Xⱼ,v>|. Under mild assumptions on X, only N=(cᵖ)ε⁻²n measurements are required, and the probability that the functional performs well is at least 1−2\exp(−cε² N).

Jan 26 - Benjamin Lees, University of Bristol

Title: The random path representation of classical spin systems

Abstract: Many models in statistical mechanics can be written in terms of a collection of (random) geometric objects. This may consist of a "soup" of random walks/loops, random currents, or random transpositions. I will introduce a model of random paths that includes several interesting models when its parameters are chosen appropriately. One such example is the spin O(N) model. Unlike similar representations of this model, the random path model is defined in terms of a product of local terms which allows many nice techniques, such as reflection positivity, to be used. I will show how to use this model to prove exponential decay of correlations for spin O(N) with an external field. The idea of the proof is very simple and applies much more generally than the alternative Lee-Yang method.

Feb 2- TBA

Title: TBA

Abstract: TBA

Feb 9- Oriane Blondel, CNRS-Lyon

Title: TBA

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Feb 16- Pietro Caputo, Roma Tre University

Title: TBA

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Feb 23- Peter Nandori, Yeshiva University (New York)

Title: TBA

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Mar 2 - Frank den Hollander, Leiden University

Title: TBA

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Mar 9 - Marta Sanz-Solé, University of Barcelona

Title: TBA

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Mar 16 - Sabine Jansen, Ludwig Maximilian University Munich

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