UNIVERSITY OF CALIFORNIARIVERSIDEStudies of Statistical-Mechanical Models Related to Quantum CodesA Dissertation submitted in partial satisfactionof the requirements for the degree ofDoctor of PhilosophyinPhysicsbyYi JiangMarch 2021Dissertation Committee:Dr. Leonid P. Pryadko, ChairpersonDr. Michael MulliganDr. Shan-Wen TsaiCopyright byYi Jiang2021The Dissertation of Yi Jiang is approved:Committee ChairpersonUniversity of California, RiversideAcknowledgmentsFirst and foremost I am grateful to my advisor, Professor Leonid Pryadko, whose guidancehelped me in all the time of my research and the completion of this dissertation. I want tothank him for his continuous support. His patience and immense knowledge are invaluablefor my academic life.I also want to thank the collaborators of this research. Part of this dissertationis a reprint of the material as it appears in “Duality and free energy analyticity boundsfor few-body Ising models with extensive homology rank”, published on May 2, 201...