# Statistical Mechanics

## Talks

Talks will be 50 mn long.

Embeddings, conformal invariance 5and universality?)

Conditional LDP for sentences and applications to disorded spatial systems.

Abstract : Cutting an i.i.d. sequence of letters into words according to an independent renewal process yields a sequence of words. We prove a "quenched" large deviation principle (LDP) for the empirical process of words, i.e., we condition on a typical letter sequence. We discuss how this problem appears via stochastic representations of the Palm distribution for disordered spatial systems like directed polymers or branching random walks in random environment.
(Based on joint work with Andreas Greven and Frank den Hollander and with Rongfeng Sun).

Parisi landscapes in finite dimensional Euclidean spaces.

Abstract : We construct a N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension the free energy of the system and overlap function are calculated exactly using the replica trick and Parisi's hierarchical ansatz. We argue that our construction is in fact valid in any finite spatial dimensions. We discuss implications of our results for the singularity spectrum describing multifractality of the associated Boltzmann-Gibbs measure. Finally we discuss several generalisations and open problems, the dynamics in such a landscape and the construction of a Generalized Multifractal Random Walk.

Cluster Areas and the Ising (Conformal) Field Theory.

Abstract : I will discuss a representation for the spin field of the d=2 critical Ising model in the scaling limit as a (conformal) random field using area measures associated with CLE (Conformal Loop Ensemble) clusters. The areas come from the scaling limit of critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. If time permits, extensions to the massive fields corresponding to off-critical scaling limits will also be discussed. The talk will be based on joint work with C. Newman and on work in progress. No knowledge of conformal field theory is needed.

Spectral measure of heavy tailed band and covariance random matrices.

Abstract : We study the asymptotic behavior of the appropiately scaled spectral measure of large random real symmetic matrices based on i.i.d. entries with common distribution in the domain of attraction of a stable law. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. This talk is based on a joint work with S. Belinschi and A. Guionnet.

Aging scaling limit for the REM-like K-process.

Abstract : K-processes are a class of Markov processes in a denumerable state space with an extra point such that when at a regular point the process jumps to the extra point (at exponential times), and starting from the extra point, the process enters finite sets of regular points with a uniform distribution over that set. These processes arise as limits of trap models in the complete graph and the hypercube. The case where the jump rates are given by the increments of a stable subordinator is associated to the REM-like trap model and also to the RHT dynamics for the REM in the complete graph and the hypercube) in time scales where those processes are close to equilibrium. In this talk, we will discuss the above points and establish a scaling limit result for the latter case (with a suitable representation and scaling) at vanishing times, related to aging.

The Fourier Spectrum of critical percolation.

Abstract : The indicator function for the existence of a percolation crossing in an n by n square can be seen as a function on the discrete cube $\{-1,1\}^E$, where $E$ is the set of edges. As such, it admits a Fourier'' expansion. In a joint work with Gabor Pete and Oded Schramm, we obtain sharp estimates for the weight'' of the Fourier coefficients at different frequencies. This good understanding of the spectrum has nice consequences for the model of dynamical percolation. For instance, we can prove in the case of the triangular lattice that the dimension of exceptional times is 31/36, and in the Z^2 case, we can prove the existence of such exceptional times.

The arcsine law as a universal aging scheme in Glauber dynamics of disordered systems.

Self-Duality and Graph Embedding.

Abstract : It is well known since Kramers and Wannier (1941) that the free energy of a lattice model at low temperature can be related to that of another model at high temperature through a duality transformation. Only a few models have been found to be self-dual, among which the Ising model on square lattice in d=2 and lattice gauge models in d=4. In both cases, this property can be related to Poincare duality applied to a partition of the embedding space R^d. This classical approach to self-duality seems to limit such models to even dimensions and a dimension-dependent structure. In this work, we construct self-dual Ising-like models in all dimensions $d\ge 2$ by relying on graph embedding duality; in other words, we consider now partitions of surfaces of high genus embedded in R^d, d>2. On one hand, we show that self-dual Ising-like models can be derived from any given self-dual graph embedding. On the other hand, we give a general method for constructing self-dual embeddings of nonplanar graphs. We show preliminary numerical results on some three dimensional examples exhibiting a first order phase transition at the Onsager point. We also recover as particular cases two Ising-like models (triangular lattice with pure triplet interactions in d=2; fcc lattice with pure quartet interactions in d=3), which were previously known to be self-dual from very different methods (Baxter 1981). We generalize these lattice models in all dimensions.

Poisson diluted ferromagnetic and spin glass models.

Abstract : We introduce a class of Poisson diluted models interpolating between pure ferromagnetic and antiferromagnetic behavior, through a spin glass phase. We address the problems of the infinite volume limit for the free energy density, the associated broken replica symmetry variational principle, the nature of the equilibrium states, and ultrametricity.

Stochastic geometry of Ising models in transverse field.

Abstract : In this talk I shall try to review stochastic geometric representations of quantum Ising models in transverse field and to explain recent results with Nick Crawford on the random current representation for these models. In particular, we derive a somewhat non-straightforward generalization of the classical switching lemma which yields a probabilistic interpretation of various truncated correlations.

The large deviation approach to non equilibrium stationary states: where are we now?

Abstract : We shall illustrate some of the insights obtained over several years in the analysis of nonequilibrium stationary states via the study of macroscopic fluctuations and discuss open problems and possible generalizations.

Abstract : For gradient models with a smooth strictly convex potential, it is known, due to Funaki and Spohn, that to each slope there corresponds a unique (tempered) ergodic infinite-volume Gibbs measure. I will report some initial results for the case of a non-convex potential. On the one hand, in our joint work with M. Biskup we describe a mechanism leading to a non-unicity, at a particular temperature, of Gibbs measures with vanishing slope. On the other hand, one still expects unicity for sufficiently small temperatures and sufficiently small slopes. I will outline the results of our joint work with S. Adams and S. Mueller, showing the first step in this direction: strict convexity of the surface tension. Our result is based on renormalization group approach introduced by D. Brydges and his collaborators.

Broadcast on a tree, reconstruction,... and glasses.

Replica Symmetry Breaking in Wireless Communication Systems.

Abstract : We analyze the performance of vector precoding in wireless communications. Mathematically speaking this comes down to a quadratic integer programming problem with an inverse Marchenko pastur kernel. We find that the replica symmetric solution is far off simulation results, while the 1RSB solution appears to be a good approximation although it still violates the positivity of entropy..

On the Nature of Isotherms at First Order Phase Transitions for Classical Lattice Models.

Abstract : For many decades the question whether there is an analytical continuation of the isotherms at at first order phase transitions remains unanswered, until Isakov gave a definite answer for the Ising model in 1984. Another way of formulating this question is : which viewpoint is correct, van der Waals' viewpoint or Mayer's viewpoint? It is known that van der Waals theory is a Mean-Field type theory. In 2003-2004 Friedli, Friedli and Pfister extended considerably the results of Isakov. More importantly they were able to study the above question for the Ising model near the Mean-Field limit. They could reconcile to some extend the viewpoints of van der Waals and Mayer. However there are still basic (and difficult) questions which remain unanswered. I shall give an overview of the mathematical results and discuss some open problems which I find important.
Original references and historical remarks on this question can be found in the expository paper Ensaios Matematicos vol 9, 1-90 (2005).

On message passing guided algorithms for solving constraint satisfaction problems.

Abstract : The statistical mechanics studies of computer science optimization problems have had two main outcomes. On the one hand they proposed a rich picture of the phase diagram of random instances, and on the other they led to new efficient heuristics for practically solving given instances (e.g. Survey Propagation). The theoretical understanding of these new algorithms is not yet completely satisfactory. In this talk I will review general ideas about message passing guided algorithms and present some results on the analysis of a representant of this class of procedures (Belief Propagation).

Quantum spin models on sparse random graphs.

Abstract : Classical spin models defined on random graphs have been the object of an intense research activity motivated, among other reasons, by their relationship to random combinatorial optimization problems. The heuristic cavity method allowed to make several qualitative and quantitative predictions about the behaviour of such random systems in their large size limit, some of these predictions having been confirmed rigorously. In this talk I will discuss a more recent development of the heuristic cavity method towards quantum models defined on random graphs. These models can be constructed, for instance, by turning a classical energy function of N Ising spins into an operator acting on the Hilbert space spanned by the 2^N configurations, and adding to it a non-commutative operator as a transverse field. Such models can be represented through path-integrals of imaginary time configurations. The cavity method can then be implemented at the quantum level by devising a sampling procedure of such spin trajectories, a procedure that can also be useful for Monte Carlo simulations.
Based on arXiv/0807.2553, joint work with F. Krzakala, A. Rosso and F. Zamponi.

Beyond the matrix-tree theorem: Fermionic (Grassmann) representation for spanning (hyper)forests and other combinatorial objects.

Abstract : Kirchhoff's matrix-tree theorem and its extensions, which express the generating polynomials of spanning trees and rooted spanning forests in a graph as determinants associated to the graph's Laplacian matrix, play a central role in electrical circuit theory and in certain exactly-soluble models in statistical mechanics. Like all determinants, those arising in Kirchhoff's theorem can be rewritten as Gaussian integrals over fermionic (Grassmann) variables.
I begin by reviewing these classic facts, and then show how the generating polynomial of unrooted spanning forests in a graph can be written as a non-Gaussian Grassmann integral involving a Gaussian term and a particular bilocal four-fermion term. Furthermore, this latter model can be mapped, to all orders in perturbation theory, onto the $N$-vector model [$O(N)$-invariant $\sigma$-model] at $N=-1$ or, equivalently, onto the $\sigma$-model taking values in the unit supersphere in ${\mathbb R}^{1|2}$ [$OSP(1|2)$-invariant $\sigma$-model]. It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free, in close analogy to (large classes of) two-dimensional $\sigma$-models and four-dimensional nonabelian gauge theories.
This is joint work with Sergio Caracciolo, Andrea Sportiello, Jesper Jacobsen, Hubert Saleur and Jesús Salas.

Disconnection of discrete cylinders and random interlacements.

Abstract : The disconnection by random walk of a discrete cylinder with a large finite connected base has been a recent object of interest. It has to do with the way paths of random walks can create interfaces. In this talk we describe some current results and explain how this problem is related to questions of percolation and to the model of random interlacements.

Fractional moments and disorder relevance for disordered pinning models.

Abstract : When one deals with disordered models, often an interesting question is to compare the quenched and annealed critical points for small disorder. Usually the correct guess is provided by the so-called (heuristic) Harris criterion. In the framework of disordered pinning models, I will present a technique, based on estimations of fractional moments of the partition function, which in various cases allows to prove that quenched and annealed critical points differ for arbitrarily small disorder (i.e. that disorder is relevant), in agreement with the Harris criterion.
Joint work with B. Derrida, G. Giacomin and H. Lacoin.

A cavity approach to Steiner problems.

Abstract : Given a graph (directed or undirected), some basic optimization problems consist in finding a sub graph with certain properties that optimizes a factorized cost function. Classical examples that fall into this class include vertex cover, perfect matching, coloring, and many others. Problems become particularly resilient to analysis when one restricts the search to subclasses of graphs that include some non-local property like connectivity, as many algorithms rely on locality. Here we shall discuss some recent advances in the design of "cavity" algorithms for non-local combinatorial optimization and statistical inference problems.

## Posters

A superconductor with 4-fermions attraction weakly perturbed by magnetic impurities.

Abstract : A paper reference is D. Borycki, Eur. Phys. J. B 65, 29–38 (2008).

Interface fluctuations for the D=1 stochastic Ginzburg-Landau equation with an asymmetric, periodic, mean-zero, on-off external potential.

Abstract : Modern biology has shown that an important number of biological processes is governed by the action of molecular complexes reminiscent in some way of macroscopic machines. The word motor'' is used for proteins that transduce at a molecular scale chemical energy into mechanical work. In the study of molecular motor, it was shown that zero-average fluctuation of a chemical potential causes net flux in appropriate conditions.
I show that the above mechanism also appears in a completely different context, the dynamic of interface in phase transition theory. To this end, I introduce a model in the framework of stochastically perturbed Ginzburg-Landau (G-L) equations for phase transitions.
I consider 1-d G-L equation in the interval $[-\epsilon^{-1},\epsilon^{-1}]$, $\eps>0$ with Neumann b.c., perturbed by an additive noise of strenght $\sqrt{\eps}$, and reaction term being the derivative of the double wells function $V(m)=m^4/4-m^2/2$. In addiction the equation is perturbed by an external field $\eps h(x,t)$ wich is an asymmetric, periodic, mean zero function in space and periodic in time. The dependence on time switches the potential on or off periodically. When $\eps=0$, the equation has equilibrium solutions that are increasing, and connect +1 with -1. I call them istantons, and I study the evolution of the solution of the perturbed equation in the limit $\eps \to 0^+$, when the initial datum is close to an istanton. I prove that, for times that may be of order of $\eps^{-1}$, the solution stays close to some instanton, whose center, suitably renormalized, converges to a Brownian plus a drift. The equation that results, in the lim! it, is the characteristic equation of on-off molecular motors, with a periodic asymmetric and mean zero on-off drift.

Hierarchical self-organisation in the Sherrington-Kirkpatrick model with multidimensional spins.

Abstract : We aim at studying the emergence of hierarchical self-organisation in mean-field spin-glasses. We consider two spin-glass models which are generated by a priori very different Gaussian correlation structures. The first model is based on a sum of hierarchically correlated random variables known as Derrida's generalised random energy model (GREM) with external field. The second model is based on an infinitesimal sum of genuinely non-hierarchically strongly correlated random variables known as the partition function of the Sherrington-Kirkpatrick (SK) model with multidimensional spins. We consider both models at the level of the free energy. The GREM with external field is also considered at a more refined level of the fluctuations of the partition function. Interestingly for the SK model with multidimensional spins, we find traces of hierarchical self-organisation at the level of the free energy in the thermodynamic limit. The hierarchical organisation of the SK model with multidimensional spins turns out to be closely related to the multidimensional generalisation of Derrida's GREM. The poster is based on joint work with Anton Bovier.

Fluctuation theory for PDMPs.

Abstract : A Piecewise Deterministic Markov Process is a random dynamical system made of deterministic motions and random jumps, with state space composed by the product of a manifold times a discrete set. The discrete variable follows a stochastic jump dynamics, while the continuous variable follows a piecewise deterministic dynamics. PDMPs are common in Applied Sciences and Ingeniering. Some examples are the modelization of biochemical processes as DNA replication genetic networks, power--stroke molecular motors. When accelerating the frequency of the chemical jumps by a factor, we prove averaging and large deviation principles. We further analyse the structure of fluctuations from the point of view of out--of--equilibrium physics.

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