A Generalized Grassmann-Pfaffian Framework for Monomer-Dimer and Spanning Trees

TL;DR

This paper introduces a unified Grassmann-Pfaffian framework for fermionic integrals, enabling new identities and computational tools for combinatorial models like monomer-dimer systems and spanning trees.

Contribution

It develops master theorems for Berezin integrals over Grassmann variables, generalizing existing identities and providing new Pfaffian and Hafnian-based representations for combinatorial problems.

Findings

Derived comprehensive identities for Berezin integrals with mixed sources

Constructed Pfaffian-sum representations for monomer-dimer systems

Presented techniques for handling singular matrices in fermionic models

Abstract

We develop a unified framework for Berezin integrals over Grassmann variables that establishes master identities for exponential quadratic fermionic forms and linear fermionic forms coupled to both bosonic and fermionic sources. The construction is rigorous for both real and complex fermions in arbitrary dimensions and remains well-defined even when the underlying matrices are singular. Our main mathematical results appear in two master theorems. Theorem 12 provides a comprehensive identity for Berezin integrals over Grassmann variables for real fermions with mixed bosonic-fermionic sources, applicable to any antisymmetric matrix. Its complex analogue, Theorem 13, yields corresponding determinant-based representations. Together, they serve as generating functionals for a wide range of combinatorial and physical models. Key applications include the dimer, monomer-dimer, matching, and almost-matching problems. We revisit the Kasteleyn theorem for planar dimers using Berezin integrals. We construct monomer-dimer systems through the \textit{Monobisyzexant (Mbsz)} function, which generalizes the Hafnian to incorporate monomer contributions and admits a Pfaffian-sum representation for planar graphs (Theorem 5); and practical techniques for handling singular matrices via unitary block decomposition (Theorem 6) and spectral analysis. We further present explicit mappings between Hafnians and Pfaffians and their submatrix generalizations (Hafnianinhos and Pfaffianinhos); an alternative source-ordered Berezin integral representation for spanning trees and forests using complex bosonic sources that regularizes the Laplacian zero mode (Theorem 10). Overall, this work offers a flexible toolkit for the theoretical analysis and computational implementation of graph-based models and lattice field theories using Berezin integrals over Grassmann variables .