Bosonic and fermionic structures in lattice models
Authors:   , Rapoport Alan 
Published: August 2024

We study the behavior of gradients squared of Gaussian fields on different graphs and their relationship with certain lattice models. In particular, we study commutative and anti-commutative squared Gaussian fields, and used them to calculate correlation functions of lattice models like the Abelian sandpile model (ASM) and uniform spanning tree (UST).

The first model to be studied is the gradient squared of the bosonic discrete Gaussian free field (dGFF) on Z^d. We first prove that this field converges to white noise in the thermodynamic limit. We also calculate its joint moments explicitly, which unveil a quasi permanental structure. We observe a similarity with the height-one field of the ASM. This is a lattice model in which every vertex on a finite grid has a value corresponding to the slope of a pile. This slope gradually increases as “grains of sand” are randomly added to the grid. When the slope surpasses a threshold the site collapses, redistributing sand to adjacent sites. When depositing grains of sand randomly, each deposition might trigger a chain reaction, impacting multiple sites. Once the system attains stationarity, we can define the height-one field as the indicator function of each site having 1 grain. In Dürre (Stoch. Process. Appl. 119(9):2725–2743, 2009) the author studies its joint cumulants in the limit, which are uncannily similar to the ones we obtained for our field, albeit with an sign of difference, having a quasi determinantal structure, instead of permanental. This similarity then begs the question: What modification of this field produces the same moments as the height-one field of the ASM?

Here is where the Grassmannian or fermionic variables come into play. If we replace the Gaussian variables by fermionic Gaussians we obtain the exact same moments expression as Dürre. What is more, our calculation method allows us to generalize the proof to any dimension, and to the triangular and hexagonal lattices in 2d, hinting towards a potential universality property. It has been conjectured that the ASM in the limit should correspond to a logarithmic CFT, an example of which is the field theory whose action is given by the gradient squared of a free fermion. We believe that our joint moments correspondence hints in that direction.

This equality poses a new question: Why? What does the fermionic GFF have to do with the ASM, so that our particular function yields the same moments as the height-one field? It is well-known that the height-one field configurations can be put in one-to-one correspondence with UST realizations. It is also known that there is a connection between UST configurations and fermionic variables. In particular, the probabilities of some edges belonging to the UST can be calculated as determinants of specific matrices, which can be expressed as expectations of products of fermionic variables. We also extend those techniques in order to find closed-form expressions of the PMF of the degree field of the UST in these lattices. To the author’s knowledge, this is the first time such expressions are given in the literature.