#### Courses

• Amine Asselah

UPEC | France

###### On some polymer models

We plan to discuss random walks in models from biology such as hydrophobic polymers. We plan to focus on folding issues, and localisation of the associated random walk. One paradigm being the Swiss Cheese picture introduced by van den Berg, Bolthausen and den Hollander in 2001.

• Christophe Garban

Université Lyon 1 | France

###### An introduction to Constructive Field Theory
The goal of this course is to give a probabilistic-oriented introduction to some of the celebrated models of Euclidean Quantum Field Theory. This field which was extremely active in the 70s/80s has seen a renewed interest in the recent years. One archetypal example of these Euclidean models is the so-called $\Phi^4_d$-model. This model can be seen as a random distribution on $\mathbb{R}^d$ whose law is informally written as follows: \begin{align*} \mu(d\phi) & \propto \exp(- \int_{\mathbb{R}^d}\{\frac 1 2 |\nabla \phi|(x)^2 + \phi^4(x)\} dx) D\phi" \\ & = \exp(- \int_{\mathbb{R}^d} \phi^4(x) dx) "\mu_{\mathrm{GFF}}(d\phi) \end{align*} where $\mu_{\mathrm{GFF}}$ is the law of a $d$-dimensional Gaussian Free Field. This course will not require any background in Field Theory, only knowledge on Brownian motion, Gaussian processes and general tools of probability theory shall be used. The (tentative) plan of the course is as follows:

1) Introduction: random fields
2) Brownian motion, the physicist's way.
3) $d$-dimensional Gaussian Free Field
4) Wick ordering ($:\phi^n :$, $:e^{\gamma \phi}:$ etc.)
5) Construction of the $\Phi^4_2$-model (dimension $d=2$)
6) Hypercontractivity and Nelson's argument
7) $\cos(\beta \phi)$ and $\exp(\gamma \phi)$ potentials
8) (time-permitting) The hierarchical $\Phi^4_3$-model
9) (time-permitting) Some singular SPDEs such as dynamical $\Phi^4_2$

• Milton Jara

IMPA | Brazil

###### Non-equilibrium Fluctuations of Interacting Particle Systems

We will give an introductory exposition to new entropy methods developed in collaboration with Patrícia Gonçalves and Otávio Menezes in order to study the large-scale behaviour of interacting particle systems. Out main application will be the derivation of the central limit theorem around its hydrodynamic limit of driven, diffusive systems, like the exclusion process in contact with reservoirs, up to dimension three.

• Claudio Landim

IMPA | Brazil

###### Metastable Markov chains

We review recent results on the metastable behavior of continuous-time Markov chains derived through the characterization of Markov chains as unique solutions of martingale problems.

Lectures are based on the paper arXiv:1807.04144

• Jason Miller

University of Cambridge | United Kingdom

###### Introduction to the Gaussian free field and Liouville quantum gravity

The purpose of this course will be to discuss various random geometries which can be generated from the Gaussian free field (GFF). In particular, we will discuss Liouville quantum gravity (LQG) which can be constructed by exponentiating the GFF. We will also discuss the level lines and flow lines of the GFF and the connection to the Schramm-Loewner evolution (SLE). Time-permitting, we will also discuss connections between LQG and random planar maps.

No knowledge of the GFF, LQG, or SLE will be assumed.

#### Talks

• Inés Armendáriz

UBA-Conicet | Argentina

###### Cycle structure of spatial random permutations

We are interested in the cycle structure of spatial random permutations that exhibit a bias towards the identity. In a first result, we show that at high enough temperature there is no condensation for a quenched system: for almost all realizations of the point process the associated measure is supported on finite cycle permutations of the set of points. Next, we propose an annealed model for the infinite volume random permutation, at both subcritical and supercritical regimes, such that infinite cycles are present only in the condensate phase, as expected. We show that the configuration of points from the random permutation coincides with the previously and independently studied physical point process.

This is joint work with P. Ferrari, N. Frevenza and S. Yuhjtman.

• Luca Avena

Leiden University | Netherlands

###### Network coarse-graining, intertwining and metastability without asymptotics

Given an irreducible Markov process on a finite but large state space, we are interested in constructing a measure-valued Markov process on a small state space describing the evolution of the distribution of the original process on a suitable time scale. This is a classical problem in the theory of metastability, where such a program is usually carried out through a proper limiting procedure (such as low temperature or large volume). Our aim is to avoid limiting procedures, for which we present a novel network coarse-graining framework based on Markov intertwining dualities. We show that, providing good'' solutions for the right intertwining equation exist, a satisfactory answer to the original goal is possible. On the other hand, for interesting examples such as a truly metastable dynamics on a large networks it turns out to be difficult (if not unfeasible) to exhibit explicit good solutions. We hence propose a randomized algorithm (based on certain spanning forests) that provides (depending on the specific model and the associated network size) approximate good'' solutions for the intertwining equation and, as a consequence, for the original problem. We conclude by discussing limitations of our approach and some open problems. Joint work with Fabienne Castell, Alex Gaudilliere and Clothilde Melot.

• Johel Beltran

Pontificia Universidad Católica del Perú | Perú

###### Coalescing random walks on a transitive graph and Kingman's coalescent

• Quentin Berger

Sorbonne Université | France

###### Directed polymers in random heavy-tail environment

The directed polymer model, introduced more than 30 years ago and intensively studied since then, can be used to describe a polymer interacting with the impurities of a heterogeneous medium. I will present a brief history of the model, and I will focus on the localization phenomenon of the trajectories, the polymer ‘’stretching’’ to reach more favorable regions of the environment. Describing the localized trajectories (super-diffusivity exponent, scaling limits, etc…) is mostly open, but I will consider the case of an environment with heavy-tail distribution, where these results are at reach. (Joint work with Niccolò Torri)

• Manuel Cabezas

Universidad Católica de Chile | Chile

###### The ant in the labyrinth

We will present scaling limits results of random walks on graph-models which belong to the same universality class as high-dimensional percolation.

• Alberto Chiarini

ETH Zurich | Switzerland

###### Entropic repulsion for the Gaussian free field conditioned on disconnection by level sets

We consider level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We derive asymptotic large deviation upper bounds on the probability that the level set of the Gaussian free field below a level $\alpha$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from an enclosing box and that the local averages of the Gaussian free field are close to a multiple of the harmonic potential of $A$.
If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the Gaussian free field experiences an entropic push down proportional to the harmonic potential of the set $A$. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice.
This talk is based on a joint work with M. Nitzschner (ETH Zurich).

• Alessandra Cipriani

Bath | UK

###### The discrete Gaussian free field on a compact manifold

In this talk we aim at defining the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a suitable random graph that replaces the square lattice $Z^d$ in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field. Joint work with Bart van Ginkel (TU Delft).

• Francis Comets

Université Paris Diderot | France

###### Rate of convergence of stochastic heat equation in weak disorder

We consider the stochastic heat equation on ${\mathbb R}^d$ with multiplicative noise given by a space-time white noise smoothed in space. As the smoothing parameter vanishes, the solution is known to converge to a random variable for $d \geq 3$ and small noise intensity. We study the rate of convergence, and the gaussian limit for the fluctuations. Joint work with C.Cosco and C.Mukherjee.

• Bernardo de Lima

Universidade Federal de Minas Gerais | Brazil

###### Truncated long-range percolation on oriented graphs

We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process. Based on joit works with Caio Alves, Aernout van Enter, Marcelo Hilário and Daniel Valesin

• Aurelia Deshayes

Univ Paris Diderot - CNRS | France

###### Front of the Friedrickson-Andersen one facilitated spin

The Friedrickson-Andersen one facilitated spin (FA-1f) model is a kinetically contrained model (KCM) where each spin can flip provided at least one nearest neighbor is empty. KCM are non attractive interacting particle systems and the study of shapes can be complicated. We study the non equilibrium dynamics of FA-1f (in dimension 1) started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for some flip parameters, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front. This is a joint work with Oriane Blondel and Cristina Toninelli.

• Mauricio Duarte

###### Reflections on Reflections

In this short talk, we will discuss general aspects of processes that include reflection in their dynamics, both in the stochastic framework and in the deterministic.

• Dirk Erhard

Universidade Federal da Bahia | Brazil

###### Scaling limit of the stochastic heat equation with exclusion interaction and asymptotics for a weakly killed random walk

This talk is about the equation $$\partial u(x,t)/\partial t = \Delta u(x,t) + [\xi(x,t)-\rho]u(x,t),\qquad x\in \mathbb Z^d, t\geq 0$$ Here, $\Delta$ is the discrete Laplacian and the $\xi$-field is a stationary and ergodic dynamic random environment with mean $\rho$ that drives the equation. I will focus on the case where $\xi$ is given in terms of a simple symmetric exclusion process, i.e., $\xi$ can be described by a field of simple random walks that move independently from each other subject to the rule that no two random walks are allowed to occupy the same site at the same time. I will discuss the behaviour of the equation when time and space are suitably scaled by some parameter $N$ that tends to infinity. It turns out that in dimension two and three a renormalisation has to be carried out in order to see a non-trivial limit. I will also give an application of our methods to study a random walk that gets weakly killed upon sharing a site with an exclusion particle. This is joint work with Martin Hairer.

• Clément Erignoux

IMPA | Brazil

###### Mathematical analysis of Motility-Induced Phase Separation and alignment phase transition in active matter models via their hydrodynamic description

(J.W. With Mourtaza Kourbane-Houssene, Julien Tailleur and Thierry Bodineau) Motility Induced Phase Separation (MIPS) and alignment phase transitions are two of the phenomena characteristic of active matter which have been the most widely studied by the physics community. However, although some progress has been achieved for mean-field models on the mathematical front, there is still a lack of mathematical understanding of many active matter models, which are by essence driven out of equilibrium by energy consumption at a microscopic model. In this talk, I will give an brief description of active matter and of the two phenomena cited above, and will show how the theory of hydrodynamic limits for particle lattice gases can, under the right scaling, allow to compute exact phase diagrams for stochastic models with purely microscopic interactions and prove the emergence of MIPS and alignment phase transition.

• Roberto Fernandez

New York University Shanghai | China

###### Relation between Gibbs and g-measures

Signals or one-dimensional spin systems can be described, in principle, both as $g$-measures or as Gibbs measures. The former were introduced in the framework of dynamical systems, while the later corresponds to a statistical mechanical setup. These formalisms are known not to be equivalent, though they admit very analogous constructions like thermodynamical functionals and equilibrium states. We discuss necessary and sufficient conditions for a $g$-measure to be Gibbs and how this relates to notions such as uniqueness and reversibility of $g$-measures.

Work in collaboration with E. Verbitskiy and S. Berghout (both from Leiden University)

• Joaquin Fontbona

University of Chile | Chile

###### Synchronization of stochastic mean-field networks of Hodgkin-Huxley neurons with noisy channels

We are interested in the collective behavior of a fully connected network of finitely many neurons when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean-field type. When the leak conductance is strictly positive, we prove that if the electrical interaction between neurons is strong enough and if the initial voltages are uniformly bounded, then the whole system synchronizes as time goes to infinity, exponentially fast and uniformly in the number of neurons, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless to any parameters values). Combining these results we deduce in the previous setting that the limiting nonlinear McKean-Vlasov PDE describing an infinite network of such neurons concentrates, as times goes to infinity, around the dynamics of a single deterministic Hodgkin-Huxley neuron (possibly with a chemical neurotransmitter channel). Our results are illustrated and complemented with numerical simulations.

Joint work with Mireille Bossy (INRIA) and Hector Olivero (Universidad de Valparaiso).

• Luiz Renato Fontes

University of São Paulo | Brazil

###### Renewal contact processes

We study the contact process with the usual exponential infection times, of rate lambda, but with general recovery times, rather than just the usual exponential recovery times. We seek conditions on the common distribution $F$ of the recovery times in order to have survival (of the infection, with positive probability) for either 1) all $\lambda > 0$; or 2) only for $\lambda$ large enough. Regarding 1), such a condition is that $F$ satisfies some regularity conditions evocative of, but going considerably beyond, inclusion in the basin of attraction of a stable law with index less than 1. And 2) holds if a) $F$ has two moments (by a standard, simple argument); or (more involvedly) if b) $F$ has a greater than 1 moment and (for a technical reason) also has a decreasing hazard rate. We will introduce the model and results in detail, and explain the main ideas and steps in our proofs. This is joint work with Domingos Marchetti, Tom Mountford and Maria Eulália Vares.

• Davide Gabrielli

DISIM University of L'Aquila | Italy

###### Large Deviations for the current of Markovian systems
Classic results for Markov processes are the celebrated Donsker-Varadhan's results on large deviations for the empirical measure and the empirical process. We discuss a result that is in-between. This is a joint large deviation principle for the empirical measure and the empirical flow that takes care of the number of jumps between the states. The corresponding rate functional is explicit. We discuss mainly applications. In particular we consider time periodic chains, particle systems and chemical reactions.

• Antonio Galves

Universidade de São Paulo | Brazil

###### Phase transition for infinite systems of spiking neurons

We consider a stochastic model for a system of spiking neurons. In this model the spiking activity of each neuron is represented by a point process having rate $1$ whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all presynaptic neurons since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to 0 and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effect leading to an abrupt loss of potential occurring at random times driven by an independent Poisson point process of rate $\gamma > 0$. For this process we prove the existence of a critical value $\gamma_c\in\, ] 0 , +\infty [$ for the leakage parameter such that the system has one or two extremal invariant measures according to whether $\gamma > \gamma_c$ or not.

This is a joint work with P. A. Ferrari, I. Grigorescu and E. Loecherbach done as an activity of the FAPESP Research, Innovation and Dissemination Center for Neuromathematics.

• Pablo Groisman

University of Buenos Aires | Argentina

###### A Model for Random Growth with Memory: Averaging Principle and Shape Theorem.

We will present a general approach to study a class of random growth models in n-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a limiting shape. We will prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain. The talk is based on joint work with A. Dembo, R. Huang and V. Sidoravicius.

• Marcelo Hilario

The Federal University of Minas Gerais - UFMG | Brazil

###### Law of Large numbers for random walks in dynamic random environment with polynomially-fast mixing

We study random walks on dynamical random environments in $1 + 1$ dimensions. Under a mild mixing assumption on the environment, we establish a law of large numbers for the random walk as well as a concentration inequality around its asymptotic speed. This mixing condition imposes a polynomial decay of covariances with sufficiently high exponent for events supported on space-time boxes separated in time. However, uniform mixing is not required. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the East model. This is a joint work with Augusto Teixeira and Oriane Blondel.

• Benedikt Jahnel

Weierstrass Institute for Applied Analysis and Stochastics | Germany

###### Percolation for Cox point processes

I will present results on continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. For this, sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization will be exhibited. Moreover, in the talk I will discuss asymptotic expressions for the percolation probability in large connection radius, large density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives. This is joint work with Christian Hirsch (LMU Munich) and Elie Calie (Orange SA Paris).

• Tobias Johnson

College of Staten Island (CUNY) | United States

###### Cover time for the frog model on trees

Place Poi$(m)$ particles at each site of a $d$-ary tree of height $n$. The particle at the root does a simple random walk. When it visits a site, it wakes up all the particles there, which start their own random walks, waking up more particles in turn. What is the cover time for this process, i.e., the time to visit every site? We show that when $m$ is large, the cover time is $O(n \log n)$ with high probability, and when $m$ is small, the cover time is at least $e^{\Omega( \sqrt{n})}$ with high probability. Both bounds are sharp by previous results of Jonathan Hermon's. This is the first result either proving that the cover time is polynomial or proving that it's nonpolymial, for any value of $m$. Joint work with Christopher Hoffman and Matthew Junge.

• Matthieu Jonckheere

UBA-Conicet | Argentina

###### Discovering independent sets of maximum size in large sparse random graphs

Finding independent set of maximum size is a NP-hard task on fixed graphs, and can take an exponentially long-time for optimal stochastic algorithms like Glauber dynamics with high activation rates. However, simple algorithms of polynomial complexity that take into account the degrees of vertices seem to perform well.
We show that the degree-greedy algorithm is indeed asymptotically optimal if a "connectivity" parameter is small enough; this parameter being strictly bigger in general that the critical parameter of the graph. This allows for instance to precisely estimate the asymptotic independence number of the sparse Erdös-Renyi graph for $p <2.5$.

• Hubert Lacoin

IMPA | Brazil

###### Wetting and Layering for discrete random interfaces

Solid-on-Solid (SOS) is a simplified surface model which has been introduced to understand the behavior of Ising interfaces in $\mathbb Z^d$ at low temperature. The simplification is obtained by considering that the interface is a graph of a function $\phi$, $\mathbb Z^{d-1} \to \mathbb Z$. In the present talk, we study the behavior of SOS surfaces in $\mathbb Z^2$ constrained to remain positive, and interacting with a potential when touching zero, corresponding to the energy functional: $$V(\phi)=\beta \sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_{x}\left( h I_{\{\phi(x)=0\}}-\infty I_{\{\phi(x)=0\}} \right).$$ We show that if $\beta$ is small enough, the system undergoes a transition from a localized phase where there is a positive fraction of contact with the wall to a delocalized one for $$h_w(\beta)= \log \left(\frac{e^{4\beta}}{e^{4\beta}-1}\right).$$ In addition by studing the free energy, we prove that the system undergoes countably many layering transitions, where the typical height of the interface jumps beween consecutive integer values.

• Florencia Leonardi

Universidade de São Paulo | Brazil

###### Some new results on the model selection problem for​ the​ Stochastic Block Model

We will present some recent advances in the estimation of the number of communities for the Stochastic Block Model. The SBM was introduced by Holland et al. (1983) and has rapidly popularised in the literature as a model for random networks exhibiting block or communit​y structure. In this model, each node in the network has associated a latent discrete random variable describing its community label, and given two nodes, the possibility of a connection between them depends only on the values of the nodes’ latent variables. We introduce​ the Krichevsky-Trofimov estimator for the number of communities given a sample of the adjacency matrix, ​without assuming a known upper bound on the true number of communities. We prove the almost sure convergence of the estimator when the sample size increases. This is a joint work with Andressa Cerqueira from Universidade de São Paulo.

• Pascal Maillard

Université Paris-Sud | France

###### 1-stable fluctuations of branching Brownian motion at critical temperature

Branching Brownian motion is a prototype of a disordered system and a toy model for spin glasses and log-correlated fields. It also has an exact duality relation with the FKPP equation, a well-known reaction diffusion equation. In this talk, I will present recent results (obtained with Michel Pain) on fluctuations of certain functionals of branching Brownian motion including the additive martingale with critical parameter and the derivative martingale. We prove non-standard central limit theorems for these quantities, with the possible limits being 1-stable laws with and asymmetry parameter depending on the functional. In particular, the derivative martingale and the additive martingale satisfy such a non-standard central limit theorem with, respectively, a totally asymmetric and a Cauchy distribution.

University of Oxford | Romania

###### SLE and Rough Paths Theory

I am going to report on some on-going research at the interface between Rough Paths Theory and Schramm-Loewner evolutions (SLE). In the study of Rough Differential Equations questions such as uniqueness/nonuniqueness of solutions depending on the behaviour of parameters of the equation appear naturally. We adapt these type of questions to the study of the backward Loewner Differential Equation in the upper half-plane. The main ideas concern the restart of the backward Loewner differential equation from the singularity. I will try to cover on the poster two parts, in which I am going to describe some general tools that lead to a better understanding of the dynamics in the closed upper half plane under the backward Loewner flow. • In the first section, I will describe some coordinate change of the Loewner equation in which we obtain in a random time change a stochastic dynamics depending only on the argument of the points. In this section, I am going to cover an analysis of this dynamics and related objects. We will further use the aforementioned estimates to reveal new information about the SLE trace at fixed capacity parametrisation times. • In the second part, I will explain how the uniqueness/non-uniqueness of solution of the backward Loewner equation started from the singularity can be understood using some property of the dynamics on the boundary. The main tool is the characterisation of boundary behaviour of Bessel processes of low and negative dimensions.

Joint work with Prof. Dmitry Belyaev and Prof. Terry Lyons. Many useful discussions with Dr. Stephen Muirhead.

Acknowledgement: I would like to acknowledge the support of ERC (Grant Agreement No.291244 Esig) 2015-2017 OMI Institute, EPSRC 1657722 2015-2018, Oxford Mathematical Department grant and EPSRC grant EP/M002896/1 (2018-2019).

• Fabio Martinelli

Universitá Roma Tre | Italy

###### On the infection time of the Duarte model: the role of energy barriers

In the Duarte model each vertex of $\mathbf Z^2$ can be either infected or healthy. In the bootstrap percolation model version, infected vertices stay infected while a healthy vertex becomes infected if at least two of its North, South, and West neighbors are infected. In the model version with kinetic constraints (KCM), each vertex with rate one changes its state by tossing a biased coin iff the same constraint is satisfied. For both versions of the model, an important problem is to determine the divergence as $q\to 0$ of the infection time of the origin when the initial infection set is q-random. For the bootstrap percolation version, the problem was solved in 2016 by B. Bollobas, H. Duminil-Copin, R. Morris, and P. Smith. For the KCM version, our recent work proves that hidden logarithmically growing energy barriers produce a much sharper divergence. The result also confirms for the Duarte model a universality conjecture for general critical KCM on $\mathbf Z^2$ put forward together with R. Morris and C. Toninelli.

Joint work with R. Morris and C. Toninelli (upper bound) and L. Mareche' and C. Toninelli (lower bound).

• Julian Martinez

###### Branching Brownian particles with spatial selection and the F-KPP equation

The F-KPP equation was introduced in 1937 as a model for the evolution of a genetic trait. This equation admits an infinite number of travelling wave solutions but only one of them has a physical meaning, the one with minimal velocity. We consider a system of $N$ interacting Branching Brownian particles and show that the empirical cumulative distribution associated to this process converges to the solution of the F-KPP equation. Additionally, for each $N$, we prove existence of a velocity for the cloud of particles. These velocities turns out to converge to the minimal one for the F-KPP, namely, a "microscopic selection principle" holds.

• Gregorio Moreno Flores

###### On random unitary operators

The Anderson model is well-known both in the mathematical physics and probability communities. However, analogous versions in the unitary framework may not be as popular among probabilists.

We will present some of these models and comment on a joint work with Olivier Bourget on dynamical localization for quantum kicked systems.

• Serguei Popov

University of Campinas | Brazil

###### On the range of two-dimensional conditioned simple random walk

We consider the two-dimensional simple random walk conditioned on never hitting the origin; strictly speaking, it is the Doob's transform of the simple random walk with respect to the potential kernel. We prove that, for "large" sets, the proportion of its sites visited by the conditioned walk is approximately a Uniform[0,1] random variable. Also, given a set that does not "surround" the origin, we prove that a.s. there are infinite linear rescalings of that set that are unvisited by the conditioned walk. These results suggest that the range of the conditioned walk has "fractal" behaviour.

• Daniel Remenik

###### The KPZ fixed point

The KPZ universality class is a broad collection of models, including directed random polymers, particle systems, random growth models and stochastic Hamilton-Jacobi equations, characterized by an unusual fluctuation behavior which is model independent but depends on the initial data, and is in some cases related to random matrix theory. A somewhat vague conjecture in the field was that there should be a universal, scaling invariant limit for all models in the KPZ class, containing all the fluctuation behavior seen in the class. In this talk I will describe joint work with K. Matetski and J. Quastel where we were able to construct and give a complete description of this limiting process, known as the KPZ fixed point. This is done by obtaining an explicit solution for the totally asymmetric simple exclusion process with arbitrary initial data which can be used naturally to pass to the limit. The limiting universal process is a Markov process evolving in a space of locally Brownian paths.

• Leonardo Rolla

UBA-Conicet | Argentina

###### Soliton decomposition of the Box-Ball System

The Box-Ball System (BBS) is a cellular automaton introduced by Takahashi and Satsuma (TS) as a discrete counterpart of the Korteweg & de Vries (KdV) differential equation. Both systems exhibit solitons, solitary waves that conserve shape and speed even after collision with other solitons. The BBS has state space $\{0,1\}^{\mathbb{Z}}$ representing boxes which may contain one ball or be empty. A carrier visits successively boxes from left to right, picking balls from occupied boxes and deposing one ball, if carried, at each visited empty box. Conservation of solitons suggests that this dynamics has many spatially-ergodic invariant measures besides the i.i.d. distribution. Building on the TS identification of solitons, we provide a soliton decomposition of the ball configurations and show that the dynamics reduces to a hierarchical translation of the components, finally obtaining an explicit recipe to construct a rich family of invariant measures. We also consider the a.s. asymptotic speed of solitons of each size.

• Santiago Saglietti

Technion | Israel

###### A strong law of large numbers for supercritical BBM with absorption
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 $L^1$. 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.

• Jaime San Martin

###### Discrete Potential Theory, some new ideas

We present some new results in Discrete Potential Theory, which contains as a principal model transient Markov chains. In particular we are interested in random walks on trees and the Markov chains induced over subsets of its state space. In particular, we show why ultrametric matrices are a main construction block in this theory.

• Avelio Sepúlveda

Université Lyon 1 | France

###### Liouville quantum gravity as a multiplicative cascade

Based on joint work with Juhan Aru and Ellen Powell. We show that Liouville quantum gravity can be seen as a generalisation of multiplicative cascades. As a consequence, the study of Liouville quantum gravity can be reduced to that of a branching random walk in which in every vertex of its genealogy has an infinite degree.

• Nahuel Soprano-Loto

Universidad de Buenos Aires | Argentina

###### Hydrodynamics of $N$ Branching Brownian motions with selection

The BBM is a system of particles performing independent Brownian motions such that at rate 1 each particle branches creating a new particle at its current site. In the $N$-BBM, only $N$ particles are kept by erasing the leftmost particle at each branching event. This process was proposed by Brunet and Derrida in the 90's and studied recently by Maillard. We show that the empirical measure of the particles at time t converges as N goes to infinity to a measure with density $u(r,t)$, the solution of a pde with a free boundary.

Joint work with Anna De Masi, Pablo Ferrari and Errico Presutti.

• Augusto Teixeira

IMPA | Brazil

• Daniel Valesin

University of Groningen | The Netherlands

###### The asymmetric multitype contact process

In the multitype contact process, vertices of a graph can be empty or occupied by a type 1 or a type 2 individual; an individual of type i dies with rate 1 and sends a descendant to a neighboring empty site with rate $\lambda_i$. We study this process on $\mathbb Z^d$ with $\lambda_1 \ge \lambda_2$ and $\lambda_1$ larger than the critical value of the (one-type) contact process. We prove that, if there is at least one type 1 individual in the initial configuration, then type 1 has a positive probability of never going extinct. Conditionally on this event, type 1 takes over a ball of radius growing linearly in time. We also completely characterize the set of stationary distributions of the process and prove that the process started from any initial configuration converges to a convex combination of distributions in this set.

• Sergio Yuhjtman

Universidad de Buenos Aires | Argentina

###### Random permutation associated to the free Bose gas in $\mathbb R^d$

Spatial random permutations are known to be related to Bose systems since 1953 (article by R.P. Feynman). Here we construct the concrete model for the infinite volume random permutation associated to the free Bose gas in $\mathbb R^d$ at both subcritical and supercritical regimes. Infinite cycles are present only in the condensate phase, as expected. The configuration of points from the random permutation coincides with the previously and independently studied physical point process. (Joint work with Pablo Ferrari and Inés Armendariz).

#### Posters

• Tulasi Ram Reddy Annapareddy

New York University Abu Dhabi | United Arab Emirates

###### Gaussian multiplicative chaos limit of the Brownian loop soup Poisson layering fields in bounded domains
We consider the primary Brownian loop soup (BLS) layering vertex fields and show the existence of the fields in smooth bounded domains for a suitable range of parameters $\beta$'s. The BLS layering vertex fields are not free, which is to say they are not the exponential of Gaussians. On the other hand, we use Weiner-Ito chaos expansion to establish that the $\lambda-\beta^2$ limit as the intensity $\lambda$ of the BLS diverges and $\beta$ goes to 0 such that $\lambda\beta^2$ is constant, is a complex Gaussian multiplicative Chaos.
This is joint work with F. Camia, A. Gandolfi and G. Peccati

• Antonio Marcos Batista do Nascimento

University of São Paulo | Brazil

###### Reaching time to equilibrium of the Metropolis dynamics for the GREM

We consider a finite state continuous-time Markov process in a random environment, namely, the Metropolis dynamics for the Generalized Random Energy Model (GREM) with a finite number of levels, and we discuss the behavior of its reaching time to equilibrium which is given by inverse of the spectral gap of its transition probability matrix. On the main result of this work, we prove the division between the system volume and the logarithm of the inverse of the gap is almost surely upper bounded by a function of the temperature that it is also the function that describe the free energy of the GREM at low temperature.

• Daniela Cuesta

Universidad de Buenos Aires | Brazil

###### Fluid limit for the coarsening phase of the condensing zero-range process

In this work we prove that the first phase of coarsening in the condensing zero range process on a finite number of sites with N particles, as N tends to infinity, is described by a fluid limit, when time is appropiately rescaled. According to this limit, in a finite time determined by the initial distribution of particles, the process reaches a state in which mass concentrates at the sites having maximal weight under the invariant measure of the underlying random walk.

• Aline Duarte

Universidade de São Paulo | Brazil

###### Estimating the interaction graph of stochastic neural dynamics

I will present a stochastic model for a system of interacting neurons and address the question of statistical model selection for this class of stochastic models. More precisely, each neuron will be modeled as a chain with memory of variable length. The relationship between a neuron and its pre and postsynaptic neurons defines an oriented graph, the interaction graph of the model. In this poster we present a consistent procedure to estimate this graph based on the observation of the spike activity of a finite set of neurons over a finite time.

• Renato Gava

Universidade Federal de São Carlos | Brazil

###### A strong invariance principle for the elephant random walk

We consider a discrete-time random walk on $\mathbb Z$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value $p_c=3/4$ where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.

• Noslen Hernandez

Universidade de São Paulo | Brazil

###### Assessment of structural learning in humans

To investigate structural learning in humans, we devised a series of experiments in which subjects were exposed to play a video-game (The goalkeeper game). The game consists in a virtual scenario in which the volunteer, playing the role of a goal keeper, has to stop the next penalty shoot by guessing the direction chosen by the kicker. During the game, stimuli are generated by stochastic chains with memory of variable length (Context tree models) and behavioral responses of participants are recorded. To test whether people exhibit structure learning and the different conjectured strategies, some statistical test are introduced. In addition, the complexity of the task is characterized using several statistical measures of complexity which are contrasted with the perception of complexity according to human performances.

• Timothy King

King's College | UK

###### Anomalous recurrence properties of random walks in Euclidean and hyperbolic space

It is well-known that the simple symmetric random walk in 3 or more dimensions is transient. However, it is possible for a zero-drift time-homogeneous Markov chain in 3 or more dimensions to be recurrent, and these walks are sometimes called anomalous. We review some known examples of anomalous walks in Euclidean space, in both the discrete and continuous time cases. We then discuss how these examples might be generalised to hyperbolic space, whose negative curvature encourages walks to be transient. In particular, we exhibit a recurrent martingale in hyperbolic space, the key idea being that, when far from the origin, the walk prefers to stay near certain one-dimensional subsets of the space. Our techniques for proving recurrence are based upon the classical work of Lamperti.

###### The contact process on fast evolving scale-free networks

We present some results on the contact process running on large scale-free networks, where nodes update their connections at independent random times. We show that depending on the parameters of the model we can observe either slow extinction for all infection rates, or fast extinction if the infection rate is small enough. This differs from previous results in the case of static scale-free networks where only the first behaviour is observed. We also show that the analysis of the asymptotic form of the metastable density of the process and its dependency on the model parameters can be used to understand the optimal mechanisms used by the infection to survive. Joint work with Peter Mörters and emmanuel Jacob.

• Guilherme Ost

Universidad Federal de Rio de Janeiro | Chile

###### Mean Field Limits for Spatially Extended Nonlinear Hawkes Processes

We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in R^d and study the associated mean field limits. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure. The limit process is described by a neural field equation. As a consequence, we provide a rigorous derivation of the neural field equation based on a thorough mean field analysis. This is a joint work with J. Chevallier, A. Duarte and E. Löcherbach.

• Ioannis Papageorgiou

Universidade de Sao Paulo | Brazil

###### Poincare type inequalities for degenerate PDMP.

We present Poincare type inequalities for a class of degenerate piecewise deterministic Markov jump processes inspired by those introduced by Galvez and Locherbach to describe the behaviour of interacting brain neurons. (Joint work with Pierre Hodara.)

• Rina Roxana Paucar Rojas

Pontificia Universidad Católica del Perú | Perú

###### Resolution of irreducible quasi ordinary surfaces

The aim of this work is to study and describe the resolution (partial and strict) of irreducible quasi ordinary surfaces (algebroids), by Lipman's approach. To achieve our goal, we define to the quasi ordinary surfaces (algebroids) and describe their parametrization by quasi ordinary branches, we also define the quasi ordinary rings, local rings of the quasi ordinary irreducible surfaces, and we study the relationship that exists between the tangent cone and singular locus (invariants that appear in these resolutions) of a quasi ordinary ring and the distinguished pairs of a quasi ordinary normalized branch that represents this ring. Also, we define the special transforms of a quasi ordinary ring and show that they are again quasi ordinary. Keywords. quasi ordinary surfaces (algebroids), resolution of singularities, blowups, quasi ordinary rings.

• Maicon Pinheiro

University of São Paulo | Brazil

###### On a continuous time random walk in Z in a dynamical random environment

Random walks in dynamical random environments have been studied under different assumptions in the last twenty years (for instance, see (Boldrighini et al., 1992), (Bandyopadhyay e Zeitouni, 2006), (Boldrighini et al., 2007), (Dolgopyat et al., 2008), (Dolgopyat e Liverani, 2009), (Yilmaz, 2009) and (Redig e Völlering , 2013)). In this work, we consider a random walk in Z in continuous time in a dynamical random environment defined by an enumerable auxiliary collection of independent and identically distributed ergodic birth-death processes; in such a way that the rates assume values in the non-compact set [1, +∞). Our overall purpose is to discuss the asymptotic behavior of the process, when time goes to +∞.

• Guilherme Reis

IMPA | Brazil

###### Interacting Diffusions on Sparse Random Graphs- Hydrodynamics and Large Deviation

Goal: to study Interacting Diffusions with interaction structure given by a sparse (inhomogeneous) random graph.
Strategy: to compare with corresponding systems with dense nonrandom interactions.
Achievement: to find the optimal sparsity in order that the two systems have the same hydrodynamic limit (McKean-Vlasov diffusions).
Background: We extend dai Pra and den Hollander proving LDPs. Other authors studied this model with different questions and results.

Authors: Roberto Imbuzeiro, Guilherme Reis

• Facundo Sapienza

Aristas | Argentina

###### Geodesics in First Passage Percolation and Distance Learning

For a random variable with values on a smooth manifold, we introduce the "Fermat distance" and its estimator, a metric that carries information about the structure of the manifold and the underlying density. The main motivation is given by applications in machine learning and statistics, such as clustering. We prove convergence of the estimator for i.i.d. samples, which involves the study of the behavior of geodesics in the context of Euclidean First Passage Percolation for non-homogeneous Poisson Point Processes.

• Federico Sau

TU Delft | Netherlands

###### Exclusion process in symmetric dynamic environment: quenched hydrodynamics

For the simple exclusion process evolving in a symmetric dynamic random environment, we derive the hydrodynamic limit from the quenched invariance principle of the corresponding random walk. For instance, if the limiting behavior of a test particle resembles that of Brownian motion on a diffusive scale, the empirical density, in the limit and suitably rescaled, evolves according to the heat equation.
Our goal is to make this connection explicit for the simple exclusion process and show how self-duality of the process enters the problem. This allows us to extend the result to other conservative particle systems (e.g. IRW, SIP) which share a similar property.
Work in progress with F. Collet, F. Redig and E. Saada.

• Tiecheng Xu

Instituto de Matemática Pura e Aplicada | Brazil

###### Stationary state of a boundary driven exclusion process with non-reversible boundary dynamics
We consider a law of large number for the empirical density of a one-dimensional, boundary driven, symmetric exclusion process. The boundary dynamics is non-reversible, and the interaction with the reservoirs depends weakly on the configuration. The proof relies on duality techniques. Joint work with C.Erignoux and C.Landim.

• Leonel Zuaznabar

Universidade de São Paulo | Brazil

###### Variation of the Random Directed Forest: convergence to the Brownian web

Several authors have studied convergence in distribution to the Brownian web under diffusive scaling of Markovian random walks. In a paper by R. Roy, K. Saha and A. Sarkar, convergence to the Brownian web is proved for a system of coalescing random paths - the Random Directed Forest- which are not Markovian. Paths in the Random Directed Forest do not cross each other before coalescence. Here we study a variation of the Random Directed Forest where paths can cross each other and prove convergence to the Brownian web. This provides an example of how the techniques to prove convergence to the Brownian web for systems allowing crossings can be applied to non-Markovian systems.