Watermelons on the half-plane

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.

arXiv comment30 pages, 4 figures, 2 tables

References in corpus (10)

• Solvable Critical Dense Polymers
• Boundary Partitions in Trees and Dimers
• Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications
• Correlation Functions of Dense Polymers and c=-2 Conformal Field Theory
• Pair correlations in sandpile model: a check of logarithmic conformal field theory
• Logarithmic conformal invariance in the Abelian sandpile model
• Logarithmic two-point correlators in the Abelian sandpile model
• Boundary correlations in planar LERW and UST
• Two-dimensional spanning webs as (1,2) logarithmic minimal model
• From elongated spanning trees to vicious random walks