Title: Statistical physics on the Z^d lattice: Proper colorings and height functions

Speaker: Ron Peled

Abstract: Statistical physics aims to understand the thermodynamic (or bulk) properties of materials from the point of view of the interactions between microscopic constituent particles. Its influence extends beyond physics to many other sciences, including social sciences, as a tool to understand the large-scale behavior of populations as it arises from the description of individuals and their local behavior. In a typical setting of (equilibrium) statistical physics, one studies the aggregate properties of spins on a large domain in a lattice, often the d-dimensional integer lattice Z^d, with the probability of each configuration of spins being proportional to exp(-1/T H), where T specifies the temperature and H the energy (or Hamiltonian) of the specific configuration as measured from the local interactions of its spins. Spin systems of this type differ from each other in the choice of allowed spin states and the function H. They exhibit quite diverse phenomena including phase transitions as the temperature varies, manifesting in qualitatively different limiting properties as the domain size tends to infinity, and the breaking of symmetries inherent in the microscopic description. The lattice dimension d often plays a significant role.

The goal of the lectures is to present various types of emergent behavior in spin systems and the methods that may be used to mathematically establish their presence. To reduce technicalities to a minimum, the course will focus on two types of spin systems, whose understanding has advanced considerably in recent years, which serve as good guides to the difficulties encountered, the diverse set of possible behaviors and the methods of rigorous analysis. No prerequisites beyond graduate level (and mostly undegraduate level) probability theory will be assumed.

Proper colorings: Consider the task of coloring the vertices of a cube in Z^d with q colors so that no two adjacent vertices are colored the same. In how many ways can this be done? Sampling such a coloring uniformly at random, does it exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depends on the dimension d and the number of colors q, focusing on disordered, critical and long-range order regimes. The model can be seen as the zero-temperature limit of the antiferromagnetic q-state Potts model. The methods used in its study, including entropy and coupling methods, Gibbs measures and their classification and refined contour analysis, apply in some generality to discrete spin systems with short-range interactions.

Height functions: Consider a random function f from a domain in Z^d to the reals, or to the integers, whose density takes the form

exp(-sum_{v, w neighbors} U(f_v - f_w))

for some potential function U. Such height functions, or random surfaces, generalize random walks to the case that the time domain is d-dimensional. How strongly do these functions fluctuate? For a given potential U and dimension d, do the fluctuations grow (delocalization) or stay bounded (localization) as the domain tends to Z^d? Real-valued random surfaces form the most basic example of systems with continuous symmetry, with the case U(x)=x^2 corresponding to the lattice Gaussian free field and being the most well understood. For the real-valued case, we explain how Mermin-Wagner type arguments can prove delocalization in two dimensions while convexity methods apply to prove localization in dimensions d>=3, highlighting the many open problems remaining. We also discuss the integer-valued case whose current theory is much more limited.

Title: Geometric Random Graphs

Speaker: Srikanth Iyer

Abstract: We begin by examining phase transition in the random connection model and a model for scale-free percolation in the continuum and discuss the ultra-small-world phenomenon. This will be followed by deriving a criterion to show Poisson approximation for Poisson functionals using the Stein’s method. As an application we derive the limiting distributions of (isolated) component counts in the sparse regime and the largest k-th nearest neighbour distances. The next couple of lectures will be used to derive the Poincare inequality using the Fock space representation for Poisson functionals. We then show how a CLT can be obtained using the concept of stabilisation along with the Poincare inequality. As an example we shall consider component counts in the k-nearest neighbour graph. The Fock space representation will then be used to derive a covariance identity which along with the Mehler’s formula yields bounds on the Wasserstein distance between the distribution of a Poisson functional and the standard normal distribution. To conclude, we shall apply the bound to show a CLT for the covered volume in the Poisson-Boolean model. All properties of a Poisson point process that we need shall be presented at the start.