Stochastic processes in Mathematical Physics

Firenze, June 19-23 2006


Giambattista Giacomin (Université Paris 7)

Random Polymer Models: Disorder and Localization Phenomena

The abstract of the mini-course is available as a PDF file

Wendelin Werner (Université Paris Sud)

Survey of recent mathematical progress in the understanding of critical 2d systems

In these lectures, I will try to survey in self-contained and accessible manner some aspects of the recent progress in the mathematical understanding of random and conformally invariant two-dimensional structures. I will in particular describe work by or in collaboration with Greg Lawler, Oded Schramm, Scott Sheffield, Stas Smirnov, and I will mention and define SLE processes, Brownian loop-soups, Gaussian Free Fields. These conformally invariant structures are supposed to (and in some cases it is proved) to appear as scaling limits of two-dimensional critical models from statistical physics. A possible reference for the course can be the notes from the lectures that I gave at Les Houches last summer (there should be some substancial overlap).

Conference talks

Gerard Ben Arous (EPF Lausanne)

Wednesday 21 at 16:00
Fractional Kinetics vs FIN diffusion as Scaling limits for trap models

Trap models are simple effective models of slow relaxation in random media. As is now well known, their scaling limit in dimension one is markovian and given by the Fontes-Isopi-Newman singular diffusion. I will report on a recent joint work with Jiri Cerny, and give their scaling limit in dimension larger or equal to two. It is drastically different, non markovian and given by the Fractional Kinetics model.

Federico Camia (Vrije Univ. Amsterdam)

Tuesday 20 at 11:30
The Continuum Scaling Limit of 2D Critical Percolation

The introduction of the random fractal curves described by Schramm-Loewner Evolutions (SLEs) has greatly deepened our understanding of the large-scale structure and fractal properties of certain two dimensional lattice models whose continuum scaling limit is known or conjectured to be conformally invariant. A particularly illuminating example is that of critical percolation, where the connection with SLE can be made rigorous and can be used to obtain various critical exponents and to describe the scaling limit of the model in terms of the collection of all its interfaces. In this talk I will review some of this recent progress, focusing on joint work with Chuck Newman.

Pierluigi Contucci (Università di Bologna)

Monday 19 at 12:15
Factorization and ultrametricity in spin glasses

We show how the property of stochastic stability and the control of the fluctuations based on thermodynamic stability imply a set of factorization properties for the spin glass models both in mean field (e.g. Sherrington-Kirkpatrick) and short range (e.g. Edwards-Anderson) case. The property of ultrametricity is discussed and numerical results for the Edwards-Anderson model illustrated.

Bernard Derrida (Ècole Normale, Paris)

Wednesday 21 at 09:00
Fluctuations of current in non-equilibrium steady states

For non-equilibrium steady states in contact with two reservoirs, one can calculate the large deviation function of the current starting from an additivity principle. It leads in particular to close expressions for all the cumulants of the current, which can be tested in some models. The validity of this additivity principle is limited by the existence of phase transitions beyond which the optimal profile to achieve some large deviations of the current become time-dependent. Work in collaboration with Thierry Bodineau.

Bertrand Duplantier (Saclay)

Wednesday 21 at 16:45
Path Crossings and Quantum Gravity

We focus on a family of exponents describing path-crossing probabilities, to which the name of Michael Aizenman is closely associated. The paths considered can be Brownian paths, paths traversing percolation clusters, or, more generally, SLE traces. With the help of an auxiliary problem on a random lattice i.e., in quantum gravity, we show how these exponents can be obtained in various ways by duality.

Luiz Renato Goncalves Fontes (IME, Sao Paulo)

Monday 19 at 11:30
K-processes, scaling limit and aging for the REM-like trap model

K-processes are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We will show how these processes arise, in a particular instance, as scaling limits of the REM-like trap model at low temperature, and that they exhibit aging behavior

Gian Michele Graf (ETH Zürich)

Tuesday 20 at 16:00
Scattering of magnetic edge states

A quantum mechanical charged particle moves in an unbounded two-dimensional domain under the influence of a homogeneous magnetic field. Under some geometric condition on the boundary the spectrum is absolutely continuous and may be interpreted in terms of edge states. We construct their scattering theory and establish its completeness. For large magnetic fields the scattering phase attains a limit which is understood in terms of the action along classical trajectories bouncing at the edge.

Abel Klein (Univ. of California, Irvine)

Thursday 22 at 12:15
Localization for Schrödinger operators with Poisson random potential

Localization for Schrödinger operators with Poisson random potential has been a longstanding open problem although Lipschitz tails, a strong indication of localization at the bottom of the spectrum, had been known since the work of Donsker and Varadhan in 1975. I will discuss my proof with Germinet and Hislop of localization for the Schrödinger operator with a Poisson random potential at the bottom of the spectrum in any dimension. We prove exponential and dynamical localization. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity.

Joel Lebowitz (Rutgers University)

Wednesday 21 at 10:30
Phase transitions and mesoscopic structures in systems with long range interactions

I will describe the statistical mechanical derivation of mesoscopic free energy functionals for systems interacting via both short range and long range (Kac) potentials. These describe the spatial structure of the different phases coexisting at a phase transitions. They give, in particular, criteria for the formation of droplets, the wetting of a demixed binary fluid and the formation of periodic structures. The behavior of systems with power-law decay of the interactions (dipole, Coulomb) will also be discussed.

Fabio Martinelli (Università di Roma Tre)

Thursday 22 at 11:30
Kinetically constrained spin models: rigorous results

Kinetically constrained spin models are lattice 0-1 spins evolving under a Glauber (or Metropolis) type of dynamics, usually reversible w.r.t. trivial Bernoulli product measure, in which a single updating at a given vertex can occur only if the current configuration around that vertex satisfies certain specific constraints. They are intensively studied in the physics literature in connection with glass and jamming transitions and are closely related to bootstrap percolation models. The simplest example is the so called Fredrickon-Andersen (FA-f) in which the spin at site x can flip only if f among its neighbors have value zero. Due to the degenaracy of the jump rates the configuration space can be broken into different irreducible components and the invariant measure is not unique. Such non uniqueness may lead to dynamical phase transition in the thermodynamic limit. The only rigorous result available so far has been obtained some years ago by Aldous and Diaconis for the East model in one dimension. In this talk I will report on a series of new results for a wide class of models in dimension greater than one obtained in collaboration with N. Cancrini (Rome), C. Roberto and C. Toninelli (Paris). The main achievements are upper bounds on the relaxation time up to the critical point, exponential decay of the so called persistence function and sharp asympotics near the critical point. Some of our findings contradict some previous conjectures based on numerical simulations. Our technique is based on a novel block dynamics which takes into account the kinetical constraints on larger and larger scales.

Errico Presutti (Roma Tor Vergata)

Wednesday 21 at 11:45
Magnetostriction effects in continuum Potts models

I will report on some preliminary results obtained in collaboration with A. De Masi, I. Merola and Y. Vignaud on a continuum version of the Potts model which show that at the appearance of a spontaneous magnetization the particles density has a jump and the typical particles distance decreases.

Jeffrey Schenker (Princeton University)

Tuesday 20 at 16:45
Eigenfunction localization for random band matrices

Lower bounds on the fluctuation of variables with local couplings can be used to prove exponential decay of correlation functions in a number of contexts. I will describe such an argument for a band matrix, with independent random entries in a band of width W, showing that suitable moments of the resolvent decay away from the diagonal with a localization length no larger than W5, implying a similar bound on the typical localization of eigenfunctions.

Scott Sheffield (Courant Institute, New York)

Thursday 22 at 09:45
Tug of war. What is it good for?

Tug of war is a two player zero game played as follows. First assign each player one of two disjoint target sets T1 and T2 in the plane, and fix a starting position x and a constant ε. Place the game token at x. At each step in the game, we toss a fair coin and allow the player who wins the coin toss to move the game token up to ε units in the direction of his or her choice. Repeat the above until the token reaches a target set T_i. The ith player is then declared the winner. Given parameters ε and x, write uε(x) for the probability that player one wins when both players play optimally. We show that as ε tends to zero, the functions uε(x) converge to the infinity harmonic function with boundary conditions 1 on T1 and 0 on T2.

This game and its variants yield many new results and simplified proofs of old results in Lipschitz extension theory, capacity theory, and the analysis of the p-Laplacian. We also discuss our original motivation for these game, which is a random turn version of Hex with conformal invariance properties.

This talk is based on joint work with Yuval Peres, Oded Schramm, and David Wilson.

Vladas Sidoravicius (IMPA)

Tuesday 20 at 12:15
TASEP with slow bond

S. R. Srinivasa Varadhan (Courant Institute, New York)

Monday 19 at 16:45
Homogenization of random Hamilton-Jacobi-Bellman equations

The abstract of the talk is available as a PDF file

Simone Warzel (Princeton University)

Wednesday 21 at 12:30
A remark on the level statistics of random operators on trees

The subject of this talk will be random Schrödinger operators on tree graphs. These operators are known to exhibit purely absolutely continuous spectrum and/or pure point spectrum depending on the energy and disorder regime. It is a widespread believe that for such operators the local fluctuations of the spectrum are related to the spectral type: whereas pure point spectrum should go along with Poisson statistics, purely absolutely continuous spectrum should produce a statistics with features similar to the random matrix case. We will examine this believe in the tree setting (This is joint work with Michael Aizenman).

Edward Waymire (Oregon State University)

Monday 19 at 16:00
A Generalized Taylor-Aris Formula and Skew Diffusion

This talk concerns the Taylor-Aris dispersion of a dilute solute concentration immersed in a highly heterogeneous fluid flow having possibly sharp interfaces (discontinuities) in the diffusion coefficient and flow velocity. The focus is two-fold:

  • (i) Calculation of the longitudinal effective dispersion coefficient,
  • (ii) sample path analysis of the underlying stochastic process governing the motion of solute particles.

Essentially complete solutions are obtained for both problems. This is joint work with Jorge Ramirez, Enrique Thomann, Roy Haggerty, and Brian Wood, and partially supported by an NSF collaborative award CMG 0327705 to Oregon State University. (To appear in SIAM Journal on Multiscale Modeling and Simulation)

Short talks

Francesco Caravenna (Universität Zürich)

Monday 19 at 15:10
Pinning models with laplacian interactions in (1+1)-dimension

We consider a random field φ: NR with Laplacian interactions of the form V(Δφ) and with in addition a delta-pinning reward for the field to touch the x-axis, that plays the role of a defect line. Denoting by ε≥0 the intensity of the pinning reward, we show that there is a phase transition at ε=εc>0 between a delocalized regime (ε≤εc), in which the field wanders away from the defect line, and a localized regime (ε>εc), in which the field sticks close to it. Using an approach based on renewal theory we extract the scaling limits of the model. In particular, we show that in the critical regime (ε=εc) the rescaled field converges in distribution toward the derivative of a symmetric stable Levy process of index 2/5 (Joint work with J.-D. Deuschel).

Beatrice De Tiliere (Universität Zürich)

Wednesday 21 at 15:01
Loops statistics in the toroidal dimer model

Consider a fixed dimer configuration (perfect matching) of a toroidal graph. Then the superposition of this and any other dimer configuration consists of loops and doubled edges. The winding number of a dimer configuration is the number of horizontal and vertical loops of the superposition. In collaboration with Cedric Boutillier, we show that the winding number of uniformly chosen dimer configurations of an increasing sequence of toroidal hexagonal graphs, converges in law to a 2-dimensional discrete gaussian random variable.

Julien Dubedat (Courant Institute)

Tuesday 20 at 15:10
Multiple SLEs

We discuss how to define several SLE (Schramm-Loewner Evolutions) interfaces simultaneously.

Alessandra Faggionato (Roma La Sapienza)

Monday 19 at 15:20
Mott variable-range random walk

Mott variable-range random walk is a random walk in a random environment used to model phonon-assisted hopping conduction in disordered solids in which the Fermi level lies in a region of strong Anderson localization. In dimension larger than 1 Mott random walk is diffusive and we give some bounds on the diffusion matrix in agreement with Mott-Efros-Shklovskii law. In dimension 1 we show that the random walk can be diffusive or subdiffusive depending on the law of the environment. In the diffusive case, we derive the qualitative behavior of the diffusion constant at low temperature.

Anne Fey-den Boer (Eurandom)

Monday 19 at 15:30
Analytical results for Zhang's sandpile model

The existing sandpile literature deals mainly with the abelian sandpile model (ASM). We treat a less known variant, Zhang's sandpile model. This model differs in two aspects from the ASM: First, additions are not discrete grains of sand, but random amounts with a uniform distribution. Second, if a site topples, it does not give one grain to each neighbor, but divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the limit of infinite volume, his model tends to behave like the ASM. This belief is supported by simulations, but so far not by analytical investigations.

We have studied analytically the stationary distributions of this model in one dimension, for several cases of the addition distribution. Our main result is on the limit of infinitely many sites, for a nontrivial case. We find that the stationary distribution in that case indeed tends to that of the ASM.

The sandpile model is used to study self-organized criticality, mainly by physicists, so rigorous analytical results are still rather rare.

Cristian Giardiná (Eurandom)

Monday 19 at 15:01
A numerical study of ultrametricity in the Edwards-Anderson model

We show the results of experiments on the ultrametric structure of overlap distribution for the Edwards-Anderson spin-glass model.

Alexis Gillett (Vrije Universiteit)

Tuesday 20 at 15:01
Dependent continuum percolation

We introduce the following dependent continuum percolation model. Consider a Poisson point process X, on Rd and connect the points to each other according to the following rule. For each x∈X, connect x to all points within distance r(x)=Σiaidi(x), where the ai are non-negative and di(x) is the distance to the i-th nearest neighbouring point of x.

The aim is to find out for which sequences of ai the model has an infinite connected component (i.e. percolates). In this talk, we give some conditions for percolation and discuss simulation results and open problems.

Tobias Kuna (Rutgers University)

Tuesday 20 at 15:30
Realizability of point processes

Point processes are describing random distributions of points. Correlation functions are characteristics of these processes. The question of realizability is to find criteria to single out functions which could appear as correlation functions of a point process. Or more restrictive which could appear for some Gibbs measures, or for Gibbs measures with a special form of the interaction, etc. The latter problem is part of the theory of classical fluids. The question of realizability was raised as an isolated problem in context of heterogeneous materials by S. Torquato and F. H. Stillinger. These problems can be seen also as a truncated moment problem w.r.t. a class of polynomials.

Francesco Mainardi (Università di Bologna)

Wednesday 21 at 15:40
Continuous time random walk and parametric subordination in fractional diffusion

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a (generally non-Markovian) diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented Lévy process, we generate and display sample paths for some special cases.

Salvador Miracle-Sole (Marseille)

Wednesday 21 at 15:10
On the wetting transition for a SOS interface

We study the solid-on-solid interface model above an horizontal wall, with an attractive interaction when the interface is in contact with the wall, at low temperatures. The system presents a sequence of layering transitions, whose levels increase with the temperature, and complete wetting above a certain value of this quantity.

Hatem Najar (IPEI, Monastir)

Wednesday 21 at 15:20
Some results on the behavior of the I.D.S of random acoustic operators

We present results on the behavior of the integrated density of states of random operators of the form Hω=∇ρω-1. We prove Lifshitz behavior for internal spectrum edges and give its asymptotic on the bottom of the spectrum.

Misja Nuyens (Vrije Universiteit)

Monday 19 at 15:40
Bak-Sneppen avalanches

A Bak-Sneppen p-avalanche on a locally finite graph G is defined as follows. Each vertex has a value, called its fitness. Initially, all fitnesses are 1, apart from the origin's fitness, which is 0. At each discrete time step, the smallest fitness and all neighbouring fitnesses are replaced by independent uniform(0,1) random variables. The p-avalanche ends when all fitnesses are greater than p.

Letting r(p) denote the number of vertices updated by a p-avalanche, we can define a critical value pc:=inf(p: P[r(p)=∞]>0). In this talk, we give bounds for pc and discuss some open problems.

Benedetto Scoppola (Roma Tor Vergata)

Wednesday 21 at 15:30
Some spin glass ideas applied to the clique problem

In this talk we introduce a new algorithm to study some NP-complete problems. This algorithm is a Markov Chain Monte Carlo (MCMC) inspired by the cavity method developed in the study of spin glass. We will focus on the maximum clique problem and we will compare this new algorithm with several standard algorithms on some DIMACS benchmark graphs and on random graphs. The performances of the new algorithm are quite surprising.

Pieter Trapman (Vrije Universiteit)

Tuesday 20 at 15:20
Exponential growth of a spatial epidemic

We consider long-range percolation on Zd as a model for infection spread. A pair of vertices at distance x is connected by an edge with probability p(x) independently of all other pairs. We consider the number of vertices in Zd at graph distance less or equal than i, say Ni, and show that for certain p(x), E(Ni) grows exponentially. We discuss whether we can obtain exponential growth with positive probability.