The inclusion process: duality and correlation inequalities

TL;DR

This paper establishes a comparison inequality between independent random walkers and the symmetric inclusion process (SIP), revealing SIP as a bosonic analogue of the fermionic symmetric exclusion process, and derives new correlation inequalities for these systems.

Contribution

It introduces a comparison inequality for SIP and related processes, demonstrating SIP's role as a bosonic counterpart to the symmetric exclusion process and deriving new correlation inequalities.

Findings

Proved a comparison inequality between independent walkers and SIP.

Established correlation inequalities for SIP and related diffusions.

Demonstrated duality and correlation inequalities for boundary-driven SIP.

Abstract

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other -- a process which we call here the symmetric inclusion process (SIP) -- or repel each other -- a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction, -- the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process, which is fermionic. Finally, we consider a boundary driven version of the SIP for which we prove duality and then obtain correlation inequalities.