Abstracts – SMMP

Stochastic Models in Mathematical Physics 5/9 - 8/9

A Conference In Memory of Dmitry (Dima) Ioffe

Marek Biskup, UCLA:

Large-scale metric properties of long-range percolation

I will discuss the asymptotic behavior of the shortest-path metric in long-range percolation models on hypercubic lattices. Starting with an overview of five different regimes of behavior depending on the values of the spatial dimension and the exponent governing the power-law decay of edge probabilities, I will describe recent results on one specific regime (namely, that when the said exponent lies between the spatial dimension and twice the spatial dimension) where sharp polylogarithmic scaling of the shortest-path distance relative to the Euclidean distance takes place. Surprisingly, the precise polylogarithmic term is modulated by a non-constant log-log-periodic function that arises from the minimizing paths having a fairly rigid dyadic hierarchical structure all the way to the lattice scales. Based on joint works with J. Lin and A. Krieger.

Thierry Bodineau, CNRS, Ecole Polytechnique (video presentation):

Large deviations for a hard sphere gas

A gas dynamics can be modelled by a billiard made of hard spheres, moving according to the laws of classical mechanics. Initially the spheres are randomly distributed according to a probability measure which is then transported by the flow of the deterministic dynamics. Since the seminal work of Lanford, it is known in the kinetic limit that the gas density converges towards the Boltzmann equation (at least for a short time). In this talk, we are going to discuss the large deviations associated with atypical densities far from the Boltzmann equation. This analysis relies on cluster expansion to study the correlations created by the Hamiltonian dynamics.

This is joint work with I. Gallagher, L. Saint-Raymond,S. Simonella

Erwin Bolthausen, University of Zurich:

A one-dimensional spin model with Kac type interaction, and a continuous symmetry

A celebrated problem in quantum mechanics concerns the effective mass of the Frohlich polaron in the strong coupling limit. Feynman gave a path integral formulation which relates it to a three dimensional Brownian motion with an attractive pair interaction of Kac type. The effective mass can be expressed as the inverse of the variance parameter of the self-interacting Brownian. There is a long standing conjecture about the asymptotic behavior of the effective mass for which Spohn gave a heuristic and plausible argument. A key property is that the interaction is shift invariant. Despite of considerable progress recently, by Mukherjee, Varadhan, Lieb, Seiring and others, the key problem is however still open.

We present a much simpler model but with a similar probabilistic structure which can be analyzed rigorously.

This is joint work with Amir Dembo.

Anton Bovier, University of Bonn:

Branching Brownian motion with self-repulsion

We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via a change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term.

This is joint work with Lisa Hartung.

Pietro Caputo, Roma Tre University (video presentation):

Rapid mixing of Gibbs samplers: Coupling, Spectral Independence, and Entropy Factorizations

We discuss some recent developments in the analysis of convergence to stationarity for the Gibbs sampler of general spin systems on arbitrary graphs. These are based on two recently introduced concepts: Spectral Independence and Block Factorization of Entropy. We show that the existence of a contractive coupling for a local Markov chain implies that the system is spectrally independent, and that if a system is spectrally independent then its entropy functional satisfies a general block factorization. As a corollary, we obtain new optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models in the so-called tree-uniqueness regime, including non-local chains such as the Swendsen-Wang dynamics. The methods apply to systems with hard constraints such as proper colorings and the hard core gas. We also discuss the entropy factorization for the uniform distribution over permutations and its role in the proof of a conjectured bound on the permanent of arbitrary matrices. Based on some recent joint works with Alexandre Bristiel, Antonio Blanca, Zongchen Chen, Daniel Parisi, Alistair Sinclair, Daniel Stefankovic, and Eric Vigoda.

Nicholas Crawford, Technion - Israel Institute of Technology:

TBA

Loren Coquille, Université Grenoble Alpes:

Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees

Statistical mechanics models on trees have been studied for almost fifty years, motivations coming from the theory of spin glasses or information theory.

In the case of the Ising model, unlike the lattice case, on the tree the behavior of the free state (corresponding to empty boundary conditions) is very rich. Indeed, the free state is extremal below the critical temperature, until the spin-glass temperature $ 0 < T_{SG} < T_c $, below which there are infinitely many pure states, as was proved by Bleher, Ruiz and Zagrebnov in 1995. Soon after, a simpler argument was provided by Dima.

I will discuss the case of ℤ-valued p-SOS-models with nearest neighbor interactions of the form |ωv−ωw|p, and ferromagnetic finite-spin models on regular trees. This includes the classical SOS-model, the discrete Gaussian and the Potts model.

In a joint work with Christof Kuelske and Arnaud Le Ny, we exhibit large sets of inhomogeneous ground state configurations, and prove that there are associated extremal inhomogeneous states at low temperature.

This constitutes a generalisation of the states initially introduced by Gandolfo, Ruiz and Shlosman in 2012 in the case of the Ising model, which play a key role in the decomposition of the Ising free state.

Amir Dembo, Stanford University:

Limit law for line ensembles of Brownian polymers with geometric area tilt

Consider non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This line ensemble, which mimics the level lines of the (2+1)D solid-on-solid model above a hard wall, was studied by Caputo, Ioffe and Wachtel. In a joint work with Eyal Lubetzky and Ofer Zeitouni, we prove the following conjecture of Caputo, Ioffe and Wachtel:

When the length of bridges, followed by the number of paths, go to infinity, the law of the top k paths converges to the same limit under both zero and most, free-like, boundary conditions.

Frank Den Hollander, Leiden University & Roman Kotecky, CTS, Charles University:

Metastability for the Widom-Rowlinson model:
I. Critical droplet (Frank den Hollander)
II. Surface tension (Roman Kotecky)

This presentation consists of two talks in tandem.
In the first talk we define the static Widom-Rowlinson model on a two dimensional finite torus. The energy of a particle configuration is determined by its halo, defined as the union of small discs centred at the positions of the particles. We discuss the metastable behaviour of a dynamic version of the model, in which particles are randomly created and annihilated as if the outside of the torus were an infinite reservoir with a given chemical potential. We start with the empty torus and are interested in the first time when the torus is fully covered by small discs. We view this as the crossover time from a ‘gas phase’ to a ‘liquid phase’. We consider the metastable regime where the temperature is low and the chemical potential is supercritical. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover.

In the second talk we give a microscopic description of the surface tension at the ”liquid/gas” coexistence line. We discuss the challenges related with the accurate determination of the surface layer and its fluctuations. We show how an analysis of the surface fluctuations leads to a derivation of the low temperature asymptotics of the surface tension with an entropic term featuring fractional power of the temperature. Our analysis relies on a variety of tools: large deviations for the volume of the halo, stability properties of isoperimetric inequalities, careful analysis of the surface tension limit and its expression in terms of the spectral radius of an appropriately chosen transfer operator.

This is a joint work with Sabine Jansen (Munich) and Elena Pulvirenti (Delft).

Jean-Dominique Deuschel, Technische Universität Berlin:

On gradient estimates of the heat kernel for random walks in time-dependent random environments

We consider a random walk among time-dependent random conductances. In recent years the long-time behaviour of this model under diffusive rescaling has been intensively studied, and -- depending on the assumptions on the law of the environment -- is fairly well understood. In this talk, I will discuss how to obtain first and second space derivatives of the annealed transition density using entropy estimates that has been developed in the time-independent setting in a paper by Benjamini, Duminil-Copin, Kozma, and Yadin (2016).

This is work in progress jointly with Jean-Dominique Deuschel (TU Berlin) and Takashi Kumagai (RIMS Kyoto).

Hugo Duminil-Copin, Université de Genève, Institut des Hautes Études Scientifique:

Scaling relations for the planar random-cluster model

We will preset the scaling relations between the critical exponents β, γ, δ, η, ν, ζ as well as α (when α ≥ 0) for the random-cluster model on the square lattice. As a key input, we show and use the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing.

Lisa Hartung, Johannes Gutenberg University of Mainz:

The speed of invasion on an advancing population

We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional coupled F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by Venegas-Ortiz et al. In that paper, it was noted that the speed of invasion by the mutant trait is faster faster when the resident population ist expanding in space compared to the speed when the resident population is already present everywhere. We use probabilistic methods, in particular the Feynman-Kac representation, to provide rigorous estimates that confirm these predictions. Based on joint work in progress with A. Bovier.

Konstantin Khanin, University of Toronto:

Coalescing Fractional Brownian Motions and the KPZ problem

We'll consider large-scale point fields which naturally appear in the context of the KPZ problem.

Such point fields are geometrical objects formed by points of mass concentration, and by shocks separating the sources of these points. We'll also introduce similarly defined point fields for the process of coalescing fractional Brownian motions (cfBM). The main aim is to present theoretical arguments and numerical evidence in support of the conjecture that statistics of these points fields have the same large-time limit.

This would indicate that two objects may, in fact, belong to the same universality class.

Takashi Kumagai, Waseda University:

Spectral dimension of simple random walk on a long-range percolation cluster

Consider the long-range percolation model on the integer lattice $ Z^d $ in which all nearest-neighbor edges are present and otherwise x and y are connected with probability $ q_{x,y}:=1−¥exp(−|x−y|^{−s}) $, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e. for $ s>d $,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $ d=1, s=2 $, where the spectral dimension is discontinuous. We present various on-diagonal heat kernel bounds, in particular, the lower bounds are derived through the application of a general technique that utilizes the translation invariance of the model. Our approach is applicable to short-range models as well.

This is a joint work with Van Hao Can (Hanoi) and David A. Croydon (Kyoto).

Ross Pinsky, Technion - Israel Institute of Technology:

Comparing the inversion statistic for distribution-biased and distribution-shifted

permutations with the geometric and the GEM distributions

Given a probability distribution $ p:=\{p_k\}_{k=1}^\infty $ on the positive integers, there are two natural ways to construct a random permutation in $ S_n $ or a random permutation of $ \mathbb{N} $ from IID samples from $ p $.

One is called the $ p $ -biased construction and the other the $ p $ -shifted construction.

In the first part of the talk we consider the case that the distribution $ p $ is the geometric distribution with parameter $ 1-q\in(0,1) $. In this case, the $ p $ -shifted random permutation has the Mallows distribution with parameter $ q $ . Let $ P_n^{b;\text{Geo}(1-q)} $ and $ P_n^{s;\text{Geo}(1-q)} $ denote the biased and the shifted distributions on $ S_n $.

The expected number of inversions of a permutation under $ P_n^{s;\text{Geo}(1-q)} $ is greater than under $ P_n^{b;\text{Geo}(1-q)} $, and under either of these distributions, a permutation tends to have many fewer inversions than it would have under the uniform distribution.

For fixed $ n $, both $ P_n^{b;\text{Geo}(1-q)} $ and $ P_n^{s;\text{Geo}(1-q)} $ converge weakly as $ q\to1 $ to the uniform distribution on $ S_n $.

We compare the biased and the shifted distributions by studying the inversion statistic under $ P_n^{b;\text{Geo}(q_n)} $ and $ P_n^{s;\text{Geo}(q_n)} $ for various rates of convergence of $ q_n $ to 1.

In the second part of the talk we consider $ p $-biased and $ p $-shifted permutations for the case that the distribution $ p $ is itself random and distributed as a GEM$ (\theta) $-distribution. In particular, in both the GEM$ (\theta) $-biased and the GEM$(\theta) $-shifted cases, the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-shifted distribution with $ \theta=\frac q{1-q} $.

This allows one to consider the GEM$ (\theta) $-shifted case as the random counterpart of the Geo$ (q) $-shifted case.

We also consider another $ p $-biased distribution with random $ p $ for which the expected number of inversions behaves asymptotically as it does under the Geo$ (1-q) $-biased case with $ \theta $ and $ q $ as above, and with $ \theta\to\infty $ and $ q\to1 $.

Jeremy Quastel, University of Toronto (video presentation):

Polynuclear growth and the Toda lattice

The polynuclear growth model is one of the most important models in the KPZ universality class. Generally it has been studied in the droplet geometry, where it is equivalent to the longest increasing subsequence of a random permutation, whose solution sparked the KPZ revolution. We study it for general initial data and show that it is an integrable Markov process sharing the key structures of the KPZ fixed point, determinantal formulas for the transition probabilities and fixed time n-point distributions governed by completely integrable equations, the non-Abelian 2D Toda lattice. Joint with Konstantin Matetski and Daniel Remenik.

Senya Shlosman, Centre de Physique Théorique, CNRS, Luminy, Marseille, Skoltech, Moscow:

The Tracy-Widom distribution and where to look for it in the Ising model

I will explain that the fluctuations of the boundaries of the flat facets of the 3D Ising crystal are governed by the Tracy-Widom distribution. Based on the joint work with Patrik Ferrari.

Balint Tóth, University of Bristol, Alfréd Rényi Institute of Mathematics Budapest:

TBA

Yvan Velenik, Université de Genève:

The Ornstein–Zernike theory: a brief history of rigorous results

I'll briefly review the history of the rigorous version of the Ornstein-Zernike theory, to the development and applications of which Dima has made many essential contributions.

Vitali Wachtel, Bielefeld University:

AR(1)-sequences with Rademacher innovations

In this talk I am going to discuss persistence probabilities for autoregressive processes. I shall primerly focus on the case when innovations have a Rademacher distribution. In this particular case one has rather interesting connection to dynamical systems in discrete time. This allows one to reduce the original problem on persistence probabilities to a first-passage problem for finite Markov chains.

Ofer Zeitouni, Weizmann Institute:

Partition function asymptotics for 2D random polymers in the sub-critical regime

Let $ W_N(\beta) = E_0 [e^{ \sum_{n=1}^N \beta\omega(n,S_n) - N\beta^2/2}] $ be the partition function of a two-dimensional directed polymer in a random environment, where $ \omega(i,x), i\in \mathbb{N}, x\in \mathbb{Z}^2 $ are i.i.d. standard normal and $ \{S_n\} $ is the path of a random walk. With $ \beta=\beta_N=\hat\beta \sqrt{\pi/\log N} $ and $ \hat \beta\in (0,1) $ (the subcritical window), $ \log W_N(\beta_N) $ is known to converge in distribution to a Gaussian law of mean $ -\lambda^2/2 $ and variance $ \lambda^2 $, with $ \lambda^2=\log (1/(1-\hat\beta^2) $ $ (\textit{Caravenna, Sun, Zygouras, Ann.\ Appl.\ Probab.\ (2017)}) $.

I will describe joint work with Clement Cosco that evaluates the moments $ E [W_N( \beta_N)^q] $ in the subcritical window, for $ q=O(\sqrt{\log N}) $. If time permits, I will discuss the relation to a log-correlated structure.