Three-leg correlations in the two component spanning tree on the upper half-plane

TL;DR

This paper analyzes the asymptotic behavior of three-path correlations in a two-component spanning tree on a 2D lattice, extending known planar results to the upper half-plane with boundary conditions.

Contribution

It provides new asymptotic formulas for correlation functions in the upper half-plane, considering boundary effects and large-distance regimes.

Findings

Asymptotics for correlations when distance r is much greater than s.

Asymptotics for correlations when s is much greater than r.

Extension of planar correlation results to the upper half-plane with boundary conditions.

Abstract

We present a detailed asymptotic analysis of correlation functions for the two component spanning tree on the two-dimensional lattice when one component contains three paths connecting vicinities of two fixed lattice sites at large distance $s$ apart. We extend the known result for correlations on the plane to the case of the upper half-plane with closed and open boundary conditions. We found asymptotics of correlations for distance $r$ from the boundary to one of the fixed lattice sites for the cases $r\gg s \gg 1$ and $s \gg r \gg 1$.