MARK KAC SEMINAR

Roughly speaking, reflection positivity (RP) of a system expresses that observables on one side of a reflection plane are positively correlated with their reflection on the other side. RP has a surprising number of interesting applications in bosonic as well as fermionic (quantum) systems. In the first part of the talk we review some of these, including chessboard estimates and infrared bounds in quantum statistical physics, and, more recently, vortex-free ground states in topologically ordered systems. Since parafermions (abelian anyons) feature prominently in topologically ordered systems, we are trying to extend RP to this setting. In the second part of the talk, we take a step in that direction. In a Z/pZ graded setting, we give a direct characterisation of RP in terms of the coupling constants in the hamiltonian. (This is joint work with Arthur Jaffe.)

Regular variation is a necessary condition for several limit laws associated with null recurrent renewal chains (such as Darling Kac and arc sine laws). Also, it is known that (positive) recurrent renewal chains satisfy stable laws if and only if the renewal distribution is regularly varying.

In joint work with Peter Kevei we obtain a Darling–Kac limit theorem along subsequences in the absence of full regular variation and determine the asymptotic behaviour of the renewal function. Also, in current work in progress with Kevei we obtain local limit theorems and in the null recurrent case, we obtain the asymptotic behaviour of the renewal sequences. (again in the absence of full regular variation.)

In joint work with Douglas Coates and Mark Holland we show that stable laws along subsequences can be established for deterministic (not necessarily Markov) processes. Such processes can be described as perturbations of maps with indifferent fixed known to give rise to return times with regularly varying tails. In the first part of the talk I will introduce the required terminology and background and in the second part I will present the results (along with the main ideas of main proofs).