Reinforced Random Walks and Spin Systems

Talk - Mathematics
Connecting random walks and spin systems are a collection of mysterious bridges known as isomorphism theorems. In the classical case, these isomorphism theorems relate the local times of simple random walks to the squares of Gaussian free fields. In this talk, I will discuss
extensions of these classical isomorphism theorems to spin systems that take values in hyperbolic and spherical geometries. Here, the corresponding random walks are no longer Markovian, but are now reinforced according to their history: in the hyperbolic case, the reinforcement is positive, giving the vertex reinforced jump process (VRJP), whereas in the spherical case, the reinforcement is negative, giving the vertex diminished jump process
(VDJP). In all three geometries -- flat, hyperbolic, and spherical -- the corresponding isomorphism theorems exist due to symmetries of the underlying spin systems, and when these symmetries are realised in more complicated spin systems, the result is a 'field reinforced random walk'. Roughly speaking, field reinforced random walks have jump rates that are mediated by a 'hidden' spin system, which itself is coupled to the local time field,
and I will also discuss isomorphism theorems in this case.