MARK KAC SEMINAR

A statistical mechanics system at a second order phase transition is characterized by a set of critical exponents describing the behavior of a few key physical quantities at or close to the transition, such as the order parameter (e.g. the magnetization in the Ising model), the correlation length, the susceptibility, or the polynomial decay of correlations of local fluctuations. These critical exponents are believed to be robust under a large class of perturbations of the microscopic Hamiltonian used to model the system, and to characterize uniquely the Euclidean field theory describing the large distance behavior of correlations. It is widely expected that this field theory is conformally invariant and can be constructed as the fixed point of a Wilsonian Renormalization Group (RG) transformation. Mathematically, there are very few cases where these expectations can be rigorously substantiated, particularly if we restric our attention to cases where the limiting Euclidean field theory is non-trivial, i.e., non-Gaussian. In this seminar I will introduce a three dimensional model of self-interacting fermions admitting a non-Gaussian RG fixed point, for which many such properties can be rigorously proven. I will describe the construction of the limiting Euclidean field theory, of the robustness of its critical exponents, and I will outline a program for proving its conformal invariance. Talks based on joint works with Vieri Mastropietro, Slava Rychkov and Giuseppe Scola.

Analogous to Kolmogorov's existence theorem for random functions, inverse limits enable existence theorems for random measures [1]. While the former is based on consistent systems of random finite marginals, inverse limits are based on coherent systems of random finite histograms. The best-known example is the Dirichlet process but various other constructions are also possible. Resulting histogram limits manifest in one of four possible phases, two of which are completely random point processes [2], and an additional two that describe non-discrete random measures. The (novel) family of Gaussian histogram limits has all four phases (depending on the choice for the parameter that fixes the covariance structure). Two examples concern the Euclidean free massless scalar field in four-dimensional space-time and energy-momentum representations. Quantization of particles as discrete points in energy-momentum space emerges from complete randomness. Regularizations (UV and IR) are part of the analysis and we consider both the infinite-momentum limit and the thermodynamic limit. Interaction Lagrangians are formulated as (sub-)martingale limits in refining histogram systems. Feynman's diagrammatic expansion is discussed, and a (convergent) strong-coupling expansion is given. A block-spin form of renormalization arises naturally from histogram coarsening, and Kadanoff effective Lagrangians are implied as restriction/conditioning of a probability measure with density. To conclude we consider ways in which these developments may influence one's perspective on quantum field theory. (Joint work with G. Meyl)