Loop-free Markov chains as determinantal point processes

We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise. Introduction Let X be a discrete space. A (simple) random point process P on X is a probability measure on the set 2 of all subsets of X. P is called determinantal if there exists a |X| × |X| matrix K with rows and columns marked by elements of X, such that for any finite Y = (y1, . . . , yn) ⊂ X one has P{X ∈ 2 | Y ⊂ X} = det[Kyiyj ]i,j=1. The matrix K is called a correlation kernel for P . A similar definition can be given for X being any reasonable space; then the measures lives on locally finite subsets of X. Determinantal point processes (with X = R) have been used in random matrix theory since early 60’s. As a separate class determinantal processes were first singled out in mid-70’s in [M] where the term fermion point processes was used. The term ‘determinantal’ was introduced at the end of 90’s in [BO] and it is now widely accepted. We refer the reader to surveys [L], [BKPV], [S3] for further references and details. Determinantal point processes have been enormously effective for studying scaling limits of interacting particle systems arising in a variety of domains of mathematics and mathematical physics including random matrices, representation theory, combinatorics, random growth models, etc. However, these processes are still considered as “exotic” — the common belief is that one needs a very special probabilistic model to observe a determinantal process. The main goal of this note is to show that determinantal point processes are much more common. More exactly, we show that for any loop-free Markov chain the induced measure on trajectories is a determinantal point process. (Note that the absence of loops is essential – otherwise trajectories cannot be viewed as subsets of the phase space.) We work in a discrete state space in order to avoid technical difficulties like trajectories that may have almost closed loops, but our construction Typeset by AMS-TEX 1