Bernoulli Variables, Classical Exclusion Processes and Free Probability

We present a new description of the known large deviation function of the classical symmetric simple exclusion process by exploiting its connection with the quantum symmetric simple exclusion processes and using tools from free probability. This may seem paradoxal as free probability usually deals with non-commutative probability while the simple exclusion process belongs to the realm of classical probability. On the way, we give a new formula for the free energy—alias the logarithm of the Laplace transform of the probability distribution—of correlated Bernoulli variables in terms of the set of their cumulants with non-coinciding indices. This latter result is obtained either by developing a combinatorial approach for cumulants of products of random variables or by borrowing techniques from Feynman graphs.