Fermionic Gaussian free field structure in the Abelian sandpile model and uniform spanning tree

TL;DR

This paper constructs a rigorous fermionic Gaussian free field representation for the Abelian sandpile model and uniform spanning tree, enabling precise computation of cumulants and demonstrating universality across different lattice types.

Contribution

It introduces a finite volume fermionic Gaussian free field representation for key fields in the Abelian sandpile and spanning tree models, linking them to logarithmic field theory.

Findings

Cumulants computed explicitly in finite volume and scaling limit

Scaling limits of cumulants agree across square and triangular lattices

Results extend to higher-dimensional hypercubic lattices

Abstract

In this paper we rigorously construct a finite volume representation for the height-one field of the Abelian sandpile model and the degree field of the uniform spanning tree in terms of the fermionic Gaussian free field. This representation can be seen as the lattice representation of a free symplectic fermion field. It allows us to compute cumulants of those fields, both in finite volume and in the scaling limit, including determining the explicit normalizing constants for fields in the corresponding logarithmic field theory. Furthermore, our results point towards universality of the height-one and degree fields, as we prove that the scaling limits of the cumulants agree (up to constants) in the square and triangular lattice. We also recover the equivalent scaling limits for the hypercubic lattice in higher dimensions, and discuss how to adapt the proofs of our results to general graphs.