The book first describes connections between some basic problems and technics of combinatorics and statistical physics. The discrete mathematics and physics terminology are related to each other. Using the established connections, some exciting activities in one field are shown from a perspective of the other field. The purpose of the book is to emphasize these interactions as a strong and successful tool. In fact, this attitude has been a strong trend in both research communities recently. It also naturally leads to many open problems, some of which seem to be basic. Hopefully, this book will help making these exciting problems attractive to advanced students and researchers.

Studies the correlation of holes in random lozenge (i.e., unit rhombus) tilings of the triangular lattice. This book analyzes the joint correlation of these triangular holes when their complement is tiled uniformly at random by lozenges.

This thesis deals with questions about sampling connected partitions, drawn from applications in the analysis of political redistricting and biostatistics. We examine this sampling problem from a complexity theory perspective, proving algorithmic lower bounds in terms of complexity classes and in terms of Markov chain mixing time lower bounds. We also investigate the implicit biases of using various distributions on connected partitions as a statistical prior, by examining the typical behavior of connected partitions. The latter investigation includes empirical work inspired by a connection with self-avoiding walks, and a theorem about bond percolation of spanning trees of grid graphs. We include two other loosely related topics. First, we investigate the relationship between the geographic compactness scores and the map projections used in redistricting, and show that under any of several popular compactness scores, any map projection will reverse the orderings of some districts. Finally, we prove an intractability theorem about uniformly sampling vertices of polytopes with bounded branch-width. Open problems are included throughout.