Date of Award
1-1-2024
Document Type
Dissertation
Degree Name
Ph.D. in Mathematics
First Advisor
Jeremy T. Clark
Second Advisor
Jeremy T. Clark
Third Advisor
Kevin Beach
Relational Format
dissertation/thesis
Abstract
We compare three models for a directed polymer in a random environment (DPRE): the rectangular lattice, the branching lattice, and the diamond hierarchical lattice. Our focus is primarily on the diamond model, in which the polymers traverse a recursively defined diamond graph with a random environment composed of random variables indexed by the vertex set. The diamond graphs are constructed iteratively according to a given branching parameter b ∈ {2, 3, 4, . . .} and segmenting parameter s ∈ {2, 3, 4, . . .}, and our main result concerns the critical case b = s. In this case, we prove a distributional limit theorem for the partition function in a limiting regime, where the growth of the system is counterbalanced with an appropriate scaling of the coupling strength of the polymer to the random environment. For context, we also provide discussion of a similar distributional limit theorem in the subcritical b < s case. This extends recent work in the critical case of analogous models where disorder variables are placed at the edges of the graphs rather than the vertices.
Recommended Citation
Lochridge, Casey Reid, "Weak Disorder Limit on Diamond Hierarchical Graphs with Vertex Disorder in the Critical Case" (2024). Electronic Theses and Dissertations. 2835.
https://egrove.olemiss.edu/etd/2835