2017/09/04 by Jeremy Clark, Clark, Jeremy
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis #math.PR
paper · pdf · doi:10.48550/arxiv.1709.01181
28 pages, 1 figure
arxiv created 2017/09/04 · openalex publication_date 2017/09/04 · arxiv updated 2017/09/06 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Diamond "lattices" are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, A and B. The construction recipe for diamond graphs depends on a branching number b∈ ℕ and a segmenting number s∈ ℕ, for which a larger value of the ratio s/b intuitively corresponds to more opportunities for intersections between two randomly chosen paths. By attaching i.i.d. random variables to the bonds of the graphs, I construct a random Gibbs measure on the set of directed paths by assigning each path an "energy" given by summing the random variables along the path. For the case b=s, I propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs, and the partition function (the normalization factor of the Gibbs measure) appears to converge in law. I prove that all of the positive integer moments of the partition function converge in this limiting regime. The motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the b<s case.