5. Perimeter
These questions concern combinatorial properties of the total boundary length of random graph partitions.-
#24 SLE for ReCom
Problem 5.1.
[Jonathan Mattingly] Is there a scaling limit for the distribution over partitions generated from running ReCom on finer and finer grids on the unit square. -
#25 Perimeter in the limit
Problem 5.2.
[Jonathan Mattingly] Fix $P_1, P_2, \dots, P_L$ partitions of the unit square into 2 pieces. Compute $$\lim_{n \to \infty} \mathbb{P}_n(P_k) := \frac{\tau_n(P_k)}{\sum_{\ell = 1}^L \tau_n(P_k)}.$$ where $\tau_n(P)$ is the number of tree-weighted partitions of the $n \times n$ grid graph that look like $P$. This fraction is set up so that the leading terms cancel, so this is really picking out the second order terms, which should be something just related to the total boundary length. -
# 29 Phase trasition for exponential weighting schemes
Problem 5.3.
[Daryl DeFord] If you are exponentially down-weighting partitions based on boundary length, at what value do you see a phase transition between compact and non-compact partitions?
Cite this as: AimPL: Mathematical foundations of sampling connected balanced graph partitions, available at http://aimpl.org/connectedbalanced.