Probability Days

Abstract:

"Stochastic lattice gases" are models of interacting particles subject to stochastic dynamics. They have been widely studied for about thirty years by both mathematicians and physicists. Their structure makes it possible to analyse them rigorously, while illustrating numerous physical phenomena: in particular, one of the main objectives is to prove rigorously the convergence of the microscopic system towards a macroscopic PDE, after rescaling in time and space (also known as the ‘hydrodynamic limit’).

In the first lecture I will give several illustrations of this convergence result thanks to the most well-known example, namely the symmetric simple exclusion process. In the second lecture, I will show how to enrich these models in order to derive some phase separation at the macroscopic level, with a free boundary that moves within the system.

Abstract:

The bulk of the random graph literature concerns models that are of an inherently static nature, in that features of the random graph at a single point in time are considered. There are strong practical motivations, however, to consider random graphs that are stochastically evolving, so as to model networks’ inherent dynamics. 

In this talk I’ll discuss a set of dynamic random graph mechanisms and their probabilistic properties. Key results cover functional diffusion limits for subgraph counts (describing the behaviour around the mean) and a sample-path large-deviation principle (describing the rare-event behaviour, thus extending the seminal result for the static case developed by Chatterjee and Varadhan).

The last part of my talk will be about estimation of the model parameters from partial information. We for instance demonstrate how the model’s underlying parameters can be estimated from just snapshots of the number of edges. We also consider settings in which particles move around on a dynamically evolving random graph, and in which the graph dynamics are inferred from the movements of the particles (i.e., not observing the graph process). 

Abstract:

Over the last 30 years, a beautiful theory of scaling limits for random trees has been established. In my first lecture, I will discuss one of the foundational examples: the Brownian continuum random tree (BCRT). Consider a uniform random labelled tree on n vertices (Cayley's formula tells us that there are n^{n-2} such labelled trees; pick one uniformly at random).  It turns out that the typical distance between two vertices in such a tree scales as \sqrt{n}, and that on rescaling we get convergence in distribution to the BCRT, which is a random fractal with fascinating properties. This result is not specific only to a uniform random labelled tree, but holds (for example) for conditioned critical branching process trees with finite offspring variance.  I aim to particularly emphasise the "line-breaking" approach to these results, which goes right back to Aldous' original work on the BCRT, but has recently been given new impetus.

In my second lecture, I will focus on a branching random walk whose genealogy is given by a conditioned branching process tree (in the domain of attraction of the BCRT). The associated discrete snake gives the collection of random walk paths from the root to each of the vertices in turn. I will discuss work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell in which we show the convergence of globally centred discrete snakes to the so-called "Brownian snake driven by a Brownian excursion".  In this work, we make essential use of the line-breaking approach outlined in the first lecture.

Abstract:

The discrete membrane model (MM) is a random interface model for separating surfaces that tend to preserve curvature. It is a very close relative of the discrete Gaussian free field (DGFF), for which instead the most likely interfaces are those preserving the mean height. However, working with the two modelspresents some key differences, in that in the MM the shape is driven by the biharmonic operator, while the DGFF is essentially a Gaussian perturbation of harmonic functions. In particular, a lot of tools (electrical networks, random walk representation of the covariance) are available for the DGFF and lack in the MM. In this talk we will review some basic properties of the MM, and we will investigate a random walk representation for the covariances of the MM and what it can bring forth in terms of its scaling limits. 

This talk is based on joint works with Alessandra Cipriani (University College, London), Biltu Dan (Indian Institute of Science) and Rounak Ray (TU, Eindhoven).

Abstract:

In the first talk I will make an quick historical survey of the rigorous results on the one-dimensional Ising model with long-range interactions. A first part will be dedicated to uniqueness of the Gibbs states (Ruelle (1968); Dobrushin (1968); Bricmont, Lebowitz & Pfister (1986)) and the regularity of the free energy when the decay of the potential is fast +enough (Dobrushin (1973) Cassandro & Olivieri (1981) and its extensions in particular Capocaccia, Campanino & Olivieri (1983).

A second part will be dedicated to the existence of phase transition starting from the Kac-Thompson conjecture (1968) the Dyson results (1969), the Frohlich \& Spencer result (1982), the Imbrie result (1982) the Aizenmann, Chayes, Chayes & Newman result on the Thouless effect (1988), Imbrie & Newman result on the Berezinsky, Kosterlitz & Thouless transition (1988).

A third part will be dedicated to present results in the phase transition regime that started with Frohlich & Spencer (1981), Cassandro, Ferrari, Merola & Presutti (2001) and its extensions in particular by Cassandro, Merola, Picco & Rosikov (2014) on the definition of an interface and its fluctuations, and on a Minlos & Sinai theorem on the phase separation problem by Cassandro, Merola & Picco (2017).

In the second talk I will review heuristic arguments that were invoked to conjecture the existence of a phase transition at low temperature in particular the Landau argument. I will present toy models where the fluctuation of interfaces and localisation of the droplet in the Minlos & Sinai theory will be explained. I will give an algorithmic definition of one-dimensional contours of Cassandro, Ferrari, Merola & Presutti.

Abstract:

Many complex systems can be viewed as a network/graph consisting of vertices (e.g., individuals) connected by edges (e.g., a friendship relation). Often one believes there is some sort of community structure, where some vertices are naturally grouped together (e.g., more densely connected between themselves than to the rest of the network). Much of the community detection literature is concentrated around methods that extract communities from a given network. Our goal in this work is different, and we attempt to understand how difficult is it to determine if a network has real communities. Furthermore, we are primarily interested in the case of small or very small communities, for which many existing results and methods are not applicable.

We cast this problem as a binary hypothesis test, where the null model corresponds to a graph without community structure, and the alternative model almost the same, but it also includes a planted community – that is, a small subset of the vertices has higher connection probability than under the null. The main question is to determine the minimal size and strength of the planted community that will allow detection. The seminal work of Arias-Castro and Verzelen tackled this problem when the null model is a homogeneous random graph. In our work, however, we consider the case where the null model is inhomogeneous, as this is somewhat closer to realistic scenarios. In particular, we present a scan test and provide conditions under which it is able to detect the presence of a small community. These results are valid for a wide variety of parameter choices. Furthermore, we show that for some parameters choices the scan test is optimal, and no other test can perform better (e.g, detect smaller or weaker planted communities). Finally, we extend this scan test to adapt to many parameters of the model when the null is a rank-1 generalized random graph.

In the first part of the talk I will describe the above formulation and ensuing results, with illustrative examples and briefly touching upon the analytical methodology. In addition, I will discuss the related problem of characterizing cliques in rank-1 random graphs, which provides some insights on the role of inhomogeneity. The second part of the talk will go deeper into more technical aspects and ensuing insights. This presentation is based on joint work with Kay Bogerd and Remco van der Hofstad (https://arxiv.org/abs/1805.01688 and ongoing work).

Abstract:

Singular stochastic partial differential equations (SSPDE) first appeared in rather special contexts like the stochastic quantization of field theories or in the problem of crystal growth, the well known KPZ equation. In the last decade these equations have been intensely studied giving rise to an important branch of mathematics possibly relevant for physics. This talk will review some aspects and open problems in the subject.

Abstract:

In the first talk I will make an quick historical survey of the rigorous results on the one-dimensional Ising model with long-range interactions. A first part will be dedicated to uniqueness of the Gibbs states (Ruelle (1968); Dobrushin (1968); Bricmont, Lebowitz & Pfister (1986)) and the regularity of the free energy when the decay of the potential is fast +enough (Dobrushin (1973) Cassandro & Olivieri (1981) and its extensions in particular Capocaccia, Campanino & Olivieri (1983).
A second part will be dedicated to the existence of phase transition starting from the Kac-Thompson conjecture (1968) the Dyson results (1969), the Frohlich \& Spencer result (1982), the Imbrie result (1982) the Aizenmann, Chayes, Chayes & Newman result on the Thouless effect (1988), Imbrie & Newman result on the Berezinsky, Kosterlitz & Thouless transition (1988).
A third part will be dedicated to present results in the phase transition regime that started with Frohlich & Spencer (1981), Cassandro, Ferrari, Merola & Presutti (2001) and its extensions in particular by Cassandro, Merola, Picco & Rosikov (2014) on the definition of an interface and its fluctuations, and on a Minlos & Sinai theorem on the phase separation problem by Cassandro, Merola & Picco (2017).

In the second talk I will review heuristic arguments that were invoked to conjecture the existence of a phase transition at low temperature in particular the Landau argument. I will present toy models where the fluctuation of interfaces and localisation of the droplet in the Minlos & Sinai theory will be explained. I will give an algorithmic definition of one-dimensional contours of Cassandro, Ferrari, Merola & Presutti.

Abstract:

The Ising model is one of the simplest statistical mechanics models that displays a phase transition. While invented by Ising and Lenz to model magnetism, for which the Ising model lives on regular lattices, it is now widely used for other real-world applications as a model for cooperative behavior and consensus between people. As such, it is natural to consider the Ising model on complex networks. Since complex networks are modelled using random graphs, this leads us to study the Ising model on random graphs. In this talk, we discuss some recent results on the stationary distribution of the Ising model on locally tree-like random graphs. We start by giving an extensive introduction to random graph models for complex networks, to set the stage of the graphs on which our Ising models live. Real-world networks tend to be highly inhomogeneous, a fact that is most prominently reflected in their degree distributions having heavy tails as described by power laws. Due to the randomness of the graphs on which the Ising model lives, there are different settings for the Ising model on it. The quenched setting describes the Ising model on the random graph as it is, while the averaged quenched setting takes the expectation w.r.t. the randomness of the graph. As such, it takes the expectation of the Boltzman distribution, which is a ratio of an exponential involving the Hamiltonian, and the partition function. In the annealed setting, the expectation is taken on both sides of the ratio. These different settings each describe different physical realities. We discuss the thermodynamic limit of the Ising model, which can be used to define the phase transition in the Ising model on locally tree-like random graphs, by describing when spontaneous magnetization exists and when not, extending work by Dembo and Montanari. We give an explicit expression for the critical value and the critical exponents for the magnetization close to it. These critical exponents depend on the power-law exponent of the degree distribution in the random graph. We also discuss central limit theorems for the total spin in the uniqueness regime, as well as a non-classical limit theorem for the total spin at the critical point in the special setting of the annealed generalized random graph. This talk is based on several joint works with Sander Dommers, Cristian Giardina, Claudio Giberti and Maria Luisa Prioriello.

Abstract:

A basic question about Markov chains is the asymptotic behavior of integrals of some function of the chain along its trajectory. In the literature, those integrals are sometimes called ‘Birkhoff averages’ or ‘additive functionals’. In the first talk, I will introduce a general strategy to estimate moment generating functions of these additive functionals in terms of the relative entropy of the chain with respect to carefully constructed reference measures. In the second talk, I will explain how to use this strategy to prove that for a large class of weakly asymmetric stochastic systems, the density of particles is well approximated by the stationary KPZ equation. This proof does not require explicit knowledge of the stationary measures of the stochastic systems, which was a major drawback of previous results.

Abstract:

It was proposed by Kardar, Parisi and Zhang in the 1980s that a large class of randomly growing interfaces exhibit universal fluctuations described mathematically by a nonlinear stochastic partial differential equation, which is now known as the Kardar-Parisi-Zhang or KPZ equation. Examples of physical systems which fall in this class are percolation of liquid in porous media, growth of bacteria colonies, currents in one dimensional traffic or liquid systems, liquid crystals etc. Surprisingly the fluctuations of such random interfaces are governed by exponents and distributions that differ from the predictions given by the classical central limit theorem and in dimension one are linked to laws emerging from random matrix theory. The link between random growth and random matrices (which still demands deeper investigation) is certain combinatorial structures. In these talks I review the current status of the field and describe some of the combinatorial structures and the advances (both older and more recent) that these have led to.