Additional Contour Properties for the Hard-Core Model on Z^2
Open Access
- Author:
- Stebbins, Daniel
- Graduate Program:
- Computer Science and Engineering (MS)
- Degree:
- Master of Science
- Document Type:
- Master Thesis
- Date of Defense:
- March 17, 2024
- Committee Members:
- Antonio Blanca, Thesis Advisor/Co-Advisor
Debarati Das, Committee Member
Chitaranjan Das, Program Head/Chair - Keywords:
- Self-Avoiding Walk
SAW
Hard-Core Model
Connective Constant
Combinatorics
Statistical Physics
Simulation
Computer Science
Lattice Gases - Abstract:
- In statistical physics, the hard-core model simulates the behavior of lattice gases with particles of non-negligible size. The size consideration is enforced by requiring particles not to be adjacent; once one vertex of the lattice is occupied, its neighbors must remain unoccupied. This rule strongly parallels the independent set problem of graph theory. The hard-core model features a parameter λ that determines the probability with which each possible configuration of occupied and unoccupied vertices arises. When λ is small, configurations with few occupied vertices are likely. These configurations are relatively chaotic and loosely packed. However, when λ is large, configurations with many occupied vertices become exponentially more likely. These configurations tend toward order, as conflicting placements waste space and reduce the number of occupied vertices. One subject of interest in this area is finding the critical value λc such that for all λ < λc the expected configuration is disordered and for all λ > λc the expected configuration is ordered. The critical value depends on the dimensions of the lattice considered. Here we consider Z2, where it is conjectured that λc ≈ 3.796. However, the rigorous bounds are still relatively loose. An upper bound on λc can be shown by bounding the number of possible contours separating disagreeing regions of a configuration, which requires formalizing the properties of said contours. It has previ- ously been shown that λc < 5.3485 using this method. This thesis presents a new contour property, explores its effect on the self-avoiding walks used for computation, and concludes with an updated bound λc < 5.1885.