The relative f-invariant and non-uniform random sofic approximations
Abstract
The -invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen in [arXiv:0802.4294], where it was used to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. In [arXiv:0902.0174] Bowen showed that the -invariant is a variant of sofic entropy; in particular it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative -invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.
Cite
@article{arxiv.2003.00663,
title = {The relative f-invariant and non-uniform random sofic approximations},
author = {Christopher Shriver},
journal= {arXiv preprint arXiv:2003.00663},
year = {2021}
}
Comments
Version 2: Theorem A has been refined, a proof that the stochastic block models are random sofic approximations has been added, other small changes