UCLA

Limit theorems are ubiquitous in probability theory. The present work samples contributions

of the author at the interface of this theory with three distinct fields: interacting particle

systems, exchangeable random variables, and long-range percolation.

In the theory of interacting particle systems, one often studies the stationary distribu

tions, which are obtained as limiting distributions of the process. We will discuss a proof

concerning characterization of these measures in the case of an attractive nearest neighbor

translation invariant spin system on the integers.

Exchangeable sequences of random variables are mixtures of i.i.d. sequences, and the

probability measure that determines the relative proportions of this mixture can be ob

tained as a limit from the exchangeable sequence itself. We will analyze the possibility of

reconstructing this probability measure from only partial information about the exchange

able sequence.

A goal in long-range percolation is to understand how chemical distance scales with

Euclidean separation. We will show that the limiting scaling behavior for a certain class of

models is polylogarithmic. This will be an improvement on existing results.