Bernoulli variables, classical exclusion processes and free probability

Abstract

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.

arXiv comment40 pages, submitted to the special issue of the Annales H. Poincare in memory of Krzysztof Gawedzki

References in corpus (13)

• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Macroscopic fluctuation theory
• Macroscopic current fluctuations in stochastic lattice gases
• Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process
• Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case
• The Exclusion Process: A paradigm for non-equilibrium behaviour
• Eigenstate Thermalization Hypothesis and Free Probability
• Microscopic versus macroscopic approaches to non-equilibrium systems
• Open Quantum Symmetric Simple Exclusion Process
• Relations between cumulants in noncommutative probability
• Can the Macroscopic Fluctuation Theory be Quantized?
• Solution to the Quantum Symmetric Simple Exclusion Process : the Continuous Case
• Cumulants of the q-semicircular law, Tutte polynomials, and heaps

Cited by in corpus (6)

• Designs via Free Probability
• Exact Entanglement in the Driven Quantum Symmetric Simple Exclusion Process
• Spacetime picture for entanglement generation in noisy fermion chains
• Large deviations of density fluctuations in the boundary driven Quantum Symmetric Simple Inclusion Process
• Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions
• Effective Action in Free Probability