Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point

Abstract

We discuss a partition-valued stochastic process in the logarithmic sector of critical cosmological topologically massive gravity. By applying results obtained in our previous works, we first show that the logarithmic sector can be modelled as an urn scheme, with a conceptual view of the random process occurring in the theory as an evolutionary process whose dynamical state space is the urn content. The urn process is then identified as the celebrated Hoppe urn model. We next show a one-to-one correspondence between Hoppe's urn model and the genus-zero Feynman diagram expansion of the log sector in terms of rooted trees. In this context, the balls in the urn model are represented by nodes in the random tree model, and the "special" ball in this P\'{o}lya-like urn construction finds a nice interpretation as the root in the recursive tree model. Furthermore, a partition-valued Markov process in which a sequence of partitions whose distribution is given by Hurwitz numbers is shown to be encoded in the log partition function. Given the bijection between the set of partitions of $n$ and the conjugacy classes of the symmetric group $S_n$, it is shown that the structure of the Markov chain consisting of a sample space that is also the set of permutations of $n$ elements, leads to a further description of the Markov chain in terms of a random walk on the symmetric group. From this perspective, a probabilistic interpretation of the logarithmic sector of the theory as a two-dimensional gauge theory on the $S_n$ group manifold is given. We suggest that a possible holographic dual to cosmological topologically massive gravity at the critical point could be a logarithmic conformal field theory that takes into account non-equilibrium phenomena.