Ajay Chandra: Constructing a massless quantum field theory over the p-adics
Abstract: In their 2003 paper Brydges, Mitter, and Scopolla studied a $\phi^4$ perturbation of a massless Gaussian scalar field over $\mathbb{R}^3$, they proved the existence of a Renormalization Group fixed point and constructed its stable manifold. I will describe further results on a simplified hierarchical version of this model. This is joint work with Abdelmalek Abdesselam and Gianluca Guadagni.Joe Chen: Spectral zeta function and quantum statistical mechanics on Sierpinski carpets
Abstract: Generalized Sierpinski carpets (GSCs) are a class of infinitely ramified fractals which include the canonical Sierpinski carpet (in 2D) and the Menger sponge (in 3D). Much progress has been made on showing the uniqueness of Brownian motion on GSC and a sharp estimate of the heat kernel trace. As a result, we can now compute the zeta function associated with the Laplacian on GSC, whose poles give the "complex dimensions" of the various spectral volumes; and show that the zeta function can be meromorphically continued to the left half plane. This allows us to compute the grand canonical partition function for any ideal gas confined to a GSC. One implication I will explain is the onset of Bose-Einstein condensation in unbounded GSC, which depends sensitively on the spectral dimension. If time permits I will also discuss the role of vacuum fluctuations (Casimir effect) in GSC.Sander Dommers: Ising models on power-law random graphs
Abstract: We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent $\tau>2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($\tau>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy. (Joint work with C. Giardina and R. van der Hofstad)Gabriel Faraud: Random walks in random environment on trees.
Abstract: The model of random walk in a random environment studied by Sinai and Solomon has been the center of much interest over the past twenty years, which has lead to numerous and precise results. In parallel to these works, many attempts have been made to extend this model to different settings. These attempts have raised many new questions, most of which still remain unanswered. We will particularly focus on the model of random walk in a random environment on trees introduced by Lyons and Pemantle in 1992. Indeed, this model presents many links with branching random walks, which makes its study easier than, for example, multidimensional RWRE. We will try to explicit these links and how we use them, and give an overview of the recent advances on this domain.Ryoki Fukushima: Localization for Brownian motion in a heavy tailed Poissonian potential
Abstract: Consider a Brownian motion evolving among a Poisson points. We introduce a repulsive interaction between the path and the Poisson points. When the interaction is of short range, Sznitman and Povel proved a certain localization result for such a process. In this talk I will present the corresponding result for the long range interaction case.Karen Hovhannisyan: Work extraction from micro canonical bath
Abstract: We determine the maximal work extractable via a cyclic Hamiltonian process from a positive-temperature ($T>0$) microcanonic state of an $N\gg 1$ particle bath. The work is much smaller than the total energy of the bath, but can be still much larger than the energy of a single bath particle, e.g. it can scale as ${\cal O}(\sqrt{N\ln N})$. Qualitatively same results are obtained for those cases, where the canonic state is unstable (e.g., due to a negative specific heat) and the microcanonic state is the only description of equilibrium. For a system attached to a microcanonic bath the concept of free energy does not {\it generally} apply, since such a system|starting from the canonic equilibrium density matrix $\rho_T$ at the bath temperature $T$|can enhance the work exracted from the microcanonic bath without changing its state $\rho_T$. This is impossible for any system attached to a canonic thermal bath due to the relation betweem the maximal work and free energy. But the concept of free energy still applies for a sufficiently large $T$. Here we find a compact expression for the {\it microcanonic free-energy} and show that in contrast to the canonic case it contains a {\it linear entropy} instead of the von Neumann entropy.Patricia Goncalves: Additive functionals of exclusion processes
Abstract: The purpose of my talk consists in establishing scaling limits of additive functionals for exclusion processes. I will consider exclusion processes denoted by (η_t)_{t>0}, evolving on the integers and starting from the Bernoulli product measure of constant parameter ρ in [0,1]. I will analyze the following functionals &Gamma_t(f) := integral over [0,t] of f(η_s) ds for proper local functions f. For f(η) := η(0), the functional Γ_t(f) is called the occupation time of the origin. I will present a method that was recently introduced in Gon\c calves and Jara (10') ``Universality of the KPZ equation'', from which we derive a local Boltzmann-Gibbs Principle for a general class of exclusion processes. For the occupation time of the origin, this principle says that the functional Γ_t(f) is very well approximated to the density of particles. As a consequence, the scaling limits of Γ_t(f) follow from the scaling limits of the density of particles. As examples I will present the symmetric simple exclusion, the mean-zero exclusion and the weakly asymmetric simple exclusion. For the latter, when the asymmetry is strong enough such that the fluctuations of the density of particles are given by the KPZ equation, we establish the limit of Γ_t(f) in terms of this solution. The case of asymmetric simple exclusion will also be discussed. This is a joint work with Milton Jara (IMPA-Brazil).Sabine Jansen: Statistical mechanics at low density and low temperature: cross-over transitions from "small" to "large" cluster sizes
Abstract: We investigate the classical statistical mechanics of particles in a continuous configuration space. Particles interact via a finite range version of a Lennard-Jones type potential. We prove that at low density and low temperature, the system displays cross-over transitions: as the temperature is lowered at constant chemical potential, the system behaves like an ideal mixture of clusters with a chemical composition (optimal cluster size) determined by the chemical potential (joint results with W. Koenig and B. Metzger). This is related to the Saha regime, or atomic and molecular limit, in quantum Coulomb system. Of particular interest is the transition from small to large cluster sizes which corresponds, in low temperature Ising models (dimension at least 2), both to a percolation and to a phase transition. I will explain how our results fit into the larger framework of phase transitions and how they relate to Mayer and virial expansions. If time permits, I will explain a connection with dynamic nucleation models (Becker-Doering equations).Anton Klimovsky: Free energy of a particle in high-dimensional Gaussian potentials with isotropic increments
Abstract: In the infinite-dimensional limit, we prove a computable saddle-point variational representation for the free energy of a particle subjected to arbitrary Gaussian random potential with isotropic increments. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.Helge Krueger: The spectrum of dynamically defined Schroedinger operators
Abstract: I will discuss recent work on understanding when the spectrum of a dynamically defined Schroedinger operator contains an interval or is a Cantor set. The main theme will be: If the parameterizing space is 1 dimensional, one gets a Cantor set, e.g. the Almost-Mathieu operator. Otherwise: Intervals.Jeffrey Kuan: Interlacing particle systems and the Gaussian free field
Abstract: We analyze the fluctuations of the height function of an interlacing particle system in the two-dimensional lattice. The main result is that these fluctuations converge to the Gaussian free field. This model has connections to the Anisotropic Kardar-Parisi-Zhang universality class and to the representation theory of Lie groups.Christof Kuelske: Metastates in Markov chain driven mean field models
Abstract: We give the construction of metastates to describe the large volume behavior in random models of statistical mechanics in the presence of phase transitions. We will then turn to mean field models subjected to external sources of randomness beyond the i.i.d. case, and discuss some fluctuation driven effects not seen in the i.i.d. case.Eunghyun Lee: Bethe ansatz solvable interacting particle systems
Abstract: We consider some exactly solvable interacting particle systems on the integer lattice : the asymmetric simple exclusion process (ASEP), the asymmetric avalanche process (ASAP) and the asymmetric zero range process (AZRP). The models are nondeterminantal, and thus the combinatorial method which was successful in the totally ASEP (TASEP) is not applicable to these models. We provide the distribution of the time-integrated current of the ASEP with the alternating initial condition by using a new combinatorial identity. Also, we provide the exact formulas of the transition probabilities for the other models. The formulas are given as a contour integral by using the Bethe ansatz and we discuss about the time-integrated current of the models.Ioan Manolescu: Universality and RSW for inhomogeneous bond percolation
Abstract: The starötriangle transformation is used to obtain an equivalence extending over the set of some (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices. Amongst the consequences are box-crossing (RSW) inequalities and the universality of alternating arms exponents (assuming they exist) for such models. The models' parameter-values are those at which the transformation is valid. This is a step towards proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.Chiranjib Mukherjee: Large deviations for Brownian intersection measures
Abstract: We consider a number of independent Brownian motions running in the $d$- dimensional Euclidean space until time $t$ and look at the spatial intersection of the paths. This intersection set carries a natural measure $L_t$, called theElena Pulvirenti: Cluster Expansion in the Canonical Ensemble
Abstract: We consider a system of particles confined in a box in d dimensions interacting via a tempered and stable potential. We prove the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime. The convergence is uniform in the volume and in the thermodynamic limit it reproduces Mayer's virial expansion providing an alternative and more direct derivation which avoids the deep combinatorial issues present in the original proof.Mykhailo Poplavskyi: Unitary Matrix Models: universality conjecture in the bulk and on the edge of the spectrum.
Abstract: We start with a formula for the joint eigenvalue distribution for the Unitary Matrix Models with some potential V. The universality conjectures for this distribution will be discussed. We show how one can reduce these conjectures to the assymptotic behavior of polynomials orthogonal on the unit circle with a varying weight and its recurrence coefficients. For the proof of universality in the bulk we obtain an integro-differential equation for the reproducing kernel of these polynomials. Using the five term recurrence relation for polynomials, that was obtained in Cantero-Moral-Velasquez papers, we prove the asymptotics for the Verblunsky coefficients and universality conjecture at the edge of the spectrum with one-interval support.David Renfrew: Fluctuations of Matrix Entries of Regular Functions of Random Matrices
Abstract: In this talk, I will discuss the fluctuations of matrix entries of regular functions of several classes of random matrices. The Gaussian case (GUE/GOE) was studied by A. Lytova and L. Pastur in 2009. Their approach significantly relies on the the unitary/ orthogonal invariance of the GUE/GOE ensemble. Using a different approach, I will explain how one can extend their results to random matrices with four finite moments on the matrix entries. I will discuss the non-universality of fluctuations for functions of Wigner matrices.Renato Santos: Law of large numbers for a transient random walk in a symmetric exclusion process
Abstract: Limit theorems for random walks in static or dynamic random environment can be obtained when the environment has good space-time mixing properties, which is not the case of the exclusion process. We discuss how to overcome this and find a regeneration structure when the walk is strongly transient.Tetiana Shcherbyna: On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble
Abstract: We consider the asymptotic of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$. We show that for the correlation function of any even order the asymptotic coincides with this for the Gaussian Unitary Ensemble up to a factor, depending only on the fourth moment of the common probability law of entries $\Im W_{jk}$, $\Re W_{jk}$, i.e. that the higher moments do not contribute to the above limit.Martin Slowik: Metastability in stochastic dynamics: Random-field Curie-Weiss-Potts model
Abstract: Metastability is a dynamical phenomenon that arises in a large variety of systems, both natural and artificial. One class of models, we are interested in, are disordered mean field spin systems at finite temperature. A particular characteristic of this kind of models is that the entropy plays a crucial role, but an exact reduction to a low-dimensional model via coarse-graining techniques is not possible. In this talk, we will see how the potential theoretic approach to metastability combined with coupling techniques can be used to compute precisely mean exit times from a metastable sets and to prove the convergence of normalized metastable exit times to an exponential distribution. We will illustrate this on the random-field Curie-Weiss-Potts model which is a simple disordered Potts spin system on the complete graph at finite temperature and with a continuous distribution of the random field.Kaspar Stucki: Multivariate Stationary Systems of Gaussian Processes
Abstract: We consider particles in the d-dimensional Euclidean space starting at the points of a Poisson point process with intensity measure Λ and moving independently of each other according to the law of some Gaussian process ξ. We describe some pairs (Λ,ξ) generating a stationary particle system.Adela Svejda: Convergence of Clock Processes in the p-spin SK Model: Extremal Case
Abstract: We consider Random Hopping Time (RHT) dynamics of mean field spin glass models and extend recent results on short time scales, obtained by Ben Arous and Gun in 2011 in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward by Bovier and Gayrard in 2010 and naturally complements their paper.Stephen Tate: Combinatorics and the Virial Expansion
Abstract: I will talk about some results and ideas from the area of Combinatorial Species and how this can be applied in understand convergence of the Virial Expansion.Alexander Vandenberg-Rodes: Negative dependence in interacting particle systems.
Abstract: We discuss the class of strong Rayleigh measures, which is naturally preserved by the symmetric exclusion process. If the state space is just the non-negative integers {0,1,...,N}, a natural generalization is to the class of measures with real-zero generating functions, and we will consider the birth-and-death chains preserving this class. This helps to characterize the interacting particle systems which enforce a strong form of negative association, at least in the mean-field situation. Joint work with T.M. Liggett.Brent Werness: Annulus Crossings of SLE, Holder Continuity, and Integration
Abstract: Various notions of regularity of SLE curves are very well understood including the almost sure Hausdorff dimension of the path and the order of Holder continuity of the path under the capacity parameterization. I will present recent work on another measure of regularity: the best order of Holder continuity obtainable under arbitrary reparametrization. With this regularity result, I will then discuss the applications of this to integration against SLE curves, and, time permitting, an application to the study of the rough path theory of SLE.Peter Windridge: Poisson-Dirichlet cycle lengths in probabilistic representations of quantum spin systems
Abstract: We consider two models of random cycles on a graph. The models were introduced by Toth and Aizenman-Nachtergaele as probabilistic representations of the Quantum Heisenberg magnet. It turns out that the physically interesting questions concern the cycle lengths, in particular whether giant cycles emerge as the graphs grows. I'll summarise existing results and explain our conjecture that the asymptotic distribution of the normalised cycle lengths is Poisson-Dirichlet, paralleling a result of Schramm. (This is ongoing work with Christina Goldschmidt and Daniel Ueltschi)Tilman Wolff: Annealed behaviour of local times in the random conductance model
Abstract: We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in Zd in the spirit of Donsker-Varadhan [DV75-83]. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.