UCLA

This thesis will examine three stochastic models using the idea of coarse-graining:

(1) A quantitative hydrodynamic limit of the Kawasaki dynamics via a "two-spatial-scale" approach, refining the original two-scale approach of Grunewald, Otto, Villani, and Westdickenberg.

(2) A quantitative ergodic theorem of the infinite-swapping process via a "two-time-scale" approach, adapted from the approach of Menz and Schlichting to the setting of an inhomogeneous diffusion.

(3) A sharp leading order asymptotic for the diameter of a long-range percolation graph via concentration inequalities, which improves a previous result by Coppersmith, Gamarnik, and Sviridenko.

A common key feature in these problems is the presence of multiple "levels" (or scales) in space or time. The solution generally involves understanding the characteristic behavior at each level and then combining the information about different levels together.