Watermelons on the half-plane

TL;DR

This paper investigates watermelon probabilities in uniform spanning forests on a semi-infinite 2D lattice, deriving universal power laws for their decay and confirming predictions from conformal field theory and Coulomb gas models.

Contribution

It provides a rigorous derivation of asymptotic decay laws for watermelon probabilities near boundaries, including non-universal constants and corrections, using advanced combinatorial and analytical methods.

Findings

Power law decay of watermelon probabilities with distance

Matching exponents with Coulomb Gas and CFT predictions

Clarification of logarithmic correction effects on decay behavior

Abstract

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb Gas technique and Conformal Field Theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.