Fermionic Discrete Gaussian Free Field (fDGFF)

• fDGFF is a lattice fermionic Gaussian theory defined via Grassmann generators and inverse Laplacian covariance, serving as a discrete bridge to logarithmic CFT.
• It employs fermionic Wick calculus and determinant formulas to derive exact cumulants for key observables like uniform spanning trees and the Abelian sandpile model.
• Planar formulations of the fDGFF yield a complete local-field theory whose scaling limits converge to symplectic fermions logCFT with central charge c = -2.

The fermionic discrete Gaussian free field (fDGFF) is a lattice fermionic Gaussian theory in which the basic variables are Grassmann generators and the covariance kernel is the inverse discrete Laplacian, i.e. the discrete Green’s function. In the 2023 graph-theoretic construction the authors use the term fermionic Gaussian free field (fGFF) and describe it as “the lattice representation of a free symplectic fermion field,” while the 2025 planar theory uses the term fermionic discrete Gaussian free field explicitly and identifies its two-dimensional scaling limit with the symplectic fermions logarithmic conformal field theory (logCFT) of central charge $c=-2$ (Chiarini et al., 2023, Adame-Carrillo et al., 5 Aug 2025). Across these works, the fDGFF is not a bosonic random field with values in $\mathbb R$, but a fermionic Gaussian state on a Grassmann algebra whose local composite observables encode the Abelian sandpile model (ASM) and the uniform spanning tree (UST).

1. Finite-volume definition and lattice formulations

The finite-volume formulation on a graph starts from a connected graph $\mathcal G=(\Lambda,E)$, typically a finite subset of $\mathbb Z^d$ or of the triangular lattice, together with Grassmann generators $\{\psi_v,\bar\psi_v:v\in\Lambda\}$. The unnormalized Dirichlet state is

$$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$

and the normalized state is obtained by dividing by $\det(-\Delta_\Lambda)$. There is also a pinned or wired version on the wired graph $\Lambda^g=\Lambda\cup\{g\}$, and a key bulk statement is that the Dirichlet and pinned functionals agree on observables $F\in\Omega^{2\Lambda}$ (Chiarini et al., 2023).

The covariance is the discrete Green’s function. In particular,

$$\langle \psi_u\bar\psi_v\rangle_\Lambda = G_\Lambda(u,v),$$

so the graph Laplacian plays the same structural role as in the bosonic DGFF, but within Berezin Gaussian calculus rather than ordinary Gaussian integration. The basic algebraic identity is the fermionic determinant formula

$\mathbb R$0

which underlies all correlation computations (Chiarini et al., 2023).

In the two-dimensional square-lattice formulation, the domain is a discrete Jordan domain $\mathbb R$1 with Dirichlet boundary condition. The Grassmann algebra is generated by two fermionic variables at each interior vertex,

$\mathbb R$2

and the action is

$\mathbb R$3

The normalized correlation functional is defined by Berezin integration against $\mathbb R$4, and the two-point function is

$\mathbb R$5

with $\mathbb R$6 the Dirichlet Green function (Adame-Carrillo et al., 5 Aug 2025).

These formulations use different notations, but both are fermionic Gaussian lattice theories with inverse-Laplacian covariance. The 2023 paper therefore treats the natural discrete object that one would describe as an fDGFF even though the acronym itself is not used there (Chiarini et al., 2023).

2. Fermionic Wick calculus and composite lattice observables

Higher correlations are governed by the fermionic Wick rule: charge-unbalanced monomials vanish, and balanced monomials are determinants of covariance matrices. In the $\mathbb R$7 notation,

$\mathbb R$8

In the graph formulation this yields, for instance,

$\mathbb R$9

so cumulants are controlled by antisymmetry and determinant expansion rather than bosonic pairing sums (Chiarini et al., 2023).

The main local bilinear introduced in the graph formulation is

$\mathcal G=(\Lambda,E)$0

with

$\mathcal G=(\Lambda,E)$1

A second local field,

$\mathcal G=(\Lambda,E)$2

appears as a nilpotent correction factor in the exact ASM representation. These are local Grassmann composites, not bosonic normal-ordered polynomials (Chiarini et al., 2023).

For cumulants of $\mathcal G=(\Lambda,E)$3, the exact finite-volume formula is a sum over cyclic permutations without fixed points: $\mathcal G=(\Lambda,E)$4 This isolates the genuinely connected fermionic contribution. For edge observables, the same determinant structure appears through transfer-current matrices and gradient bilinears of the form

$\mathcal G=(\Lambda,E)$5

which makes the UST connection exact at finite volume (Chiarini et al., 2023).

The contrast with the bosonic DGFF is structural: commuting real variables are replaced by anticommuting Grassmann pairs, Gaussian measures by Berezin integrals, bosonic Wick pairings by determinant contractions, and the partition function carries a determinant in the numerator rather than a negative power of a determinant (Chiarini et al., 2023).

3. Local fields, null fields, and symmetry algebra

The 2025 work develops a full local-field theory for the planar fDGFF. One begins with the Grassmann algebra of formal field polynomials

$\mathcal G=(\Lambda,E)$6

and defines evaluation at a lattice point $\mathcal G=(\Lambda,E)$7 by translation, with the convention that variables outside the domain evaluate to zero. A field polynomial is null if all sufficiently separated correlation functions with that insertion vanish. The space of local fields is then the quotient

$\mathcal G=(\Lambda,E)$8

This quotient is the discrete analogue of the continuum state-field space (Adame-Carrillo et al., 5 Aug 2025).

Distinguished classes in the quotient are

$\mathcal G=(\Lambda,E)$9

Here $\mathbb Z^d$0 is the identity field, $\mathbb Z^d$1 is its logarithmic partner, and $\mathbb Z^d$2 are the ground fermions. Discrete Laplacians of the basic fermions are null, and the same is true for their $\mathbb Z^d$3 descendants through the factorization $\mathbb Z^d$4 (Adame-Carrillo et al., 5 Aug 2025).

A central step is the construction of current modes by discrete contour integrals against discrete monomials $\mathbb Z^d$5. The resulting operators satisfy the symplectic-fermion anticommutation relations

$\mathbb Z^d$6

with all other anticommutators vanishing. Via discrete Sugawara formulas, these modes generate commuting holomorphic and antiholomorphic Virasoro actions with central charge

$\mathbb Z^d$7

The local-field space is then identified with the logarithmic Fock space of symplectic fermions (Adame-Carrillo et al., 5 Aug 2025).

The logarithmic character is encoded in the non-diagonalizability of $\mathbb Z^d$8. In particular,

$\mathbb Z^d$9

so $\{\psi_v,\bar\psi_v:v\in\Lambda\}$0 belongs to a rank-2 Jordan block rather than being an ordinary scaling eigenfield. This is the discrete algebraic origin of logarithmic renormalization in the scaling limit (Adame-Carrillo et al., 5 Aug 2025).

4. Scaling limit and symplectic-fermion logCFT

The planar scaling-limit theorem identifies the fDGFF local-field theory with the symplectic fermions logCFT in simply connected domains. If $\{\psi_v,\bar\psi_v:v\in\Lambda\}$1 in the Carathéodory sense and $\{\psi_v,\bar\psi_v:v\in\Lambda\}$2 represent appropriately renormalized lattice fields corresponding to continuum fields $\{\psi_v,\bar\psi_v:v\in\Lambda\}$3 of generalized scaling dimensions $\{\psi_v,\bar\psi_v:v\in\Lambda\}$4, then

$\{\psi_v,\bar\psi_v:v\in\Lambda\}$5

uniformly on compact subsets of the configuration space of pairwise distinct points (Adame-Carrillo et al., 5 Aug 2025).

For genuine eigenfields of $\{\psi_v,\bar\psi_v:v\in\Lambda\}$6, a pure power renormalization suffices. For generalized eigenfields one must add a logarithmic counterterm: $\{\psi_v,\bar\psi_v:v\in\Lambda\}$7 The standard example is the logarithmic partner $\{\psi_v,\bar\psi_v:v\in\Lambda\}$8, represented discretely by

$\{\psi_v,\bar\psi_v:v\in\Lambda\}$9

whose expectation converges to

$$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$0

with $$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$1 (Adame-Carrillo et al., 5 Aug 2025).

The 2023 graph-theoretic results already establish the symplectic-fermion structure at the level of cumulants for the key observables $$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$2 and $$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$3. Their scaling limits are expressed through derivatives of the continuum harmonic Green’s function $$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$4, and in $$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$5 the ASM normalization constant is

$$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$6

The paper explicitly relates the continuum field to the symplectic-fermion bilinear

$$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$7

and proves that the lattice cumulants converge to the predicted Green-derivative formulas with the correct constant in the square lattice case (Chiarini et al., 2023).

5. Uniform spanning tree and Abelian sandpile representations

One of the main roles of the fDGFF is to represent natural observables in UST and ASM. For the UST, if $$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$8 is a set of edges and

$$\mathbb E_\Lambda^{\mathrm d}[F] = \left(\prod_{v\in\Lambda}\partial_{\bar\psi_v}\partial_{\psi_v}\right) \exp\!\left(\langle \bm\psi,-\Delta_\Lambda\bm{\bar\psi}\rangle\right)F,$$9

then

$\det(-\Delta_\Lambda)$0

At the level of vertex observables, the normalized degree field $\det(-\Delta_\Lambda)$1 satisfies

$\det(-\Delta_\Lambda)$2

so the UST degree field is represented directly by the fermionic bilinear $\det(-\Delta_\Lambda)$3 (Chiarini et al., 2023).

For the ASM, the exact finite-volume identity is

$\det(-\Delta_\Lambda)$4

where $\det(-\Delta_\Lambda)$5 is the height-one indicator. Thus the height-one field is represented not by $\det(-\Delta_\Lambda)$6 alone but by the composite $\det(-\Delta_\Lambda)$7, with $\det(-\Delta_\Lambda)$8 acting as a lattice correction factor that becomes a multiplicative constant in the scaling limit (Chiarini et al., 2023).

The 2025 local-field theory sharpens this interpretation. The centered UST degree field corresponds to a local field $\det(-\Delta_\Lambda)$9 whose leading term is

$\Lambda^g=\Lambda\cup\{g\}$0

while the centered ASM height-one field corresponds to a local field $\Lambda^g=\Lambda\cup\{g\}$1 with leading term

$\Lambda^g=\Lambda\cup\{g\}$2

The dissipation field is

$\Lambda^g=\Lambda\cup\{g\}$3

which is

$\Lambda^g=\Lambda\cup\{g\}$4

in the local-field quotient, so dissipation is logarithmic. The same work proves exact fDGFF representations for mixed ASM correlations involving height-one and dissipation insertions (Adame-Carrillo et al., 5 Aug 2025).

The 2025 paper also states explicitly that the broader claim that the full ASM scaling limit is symplectic fermions is not asserted; what is proved is the correspondence for certain local observables and their scaling limits (Adame-Carrillo et al., 5 Aug 2025).

6. Universality, scope, and relation to the bosonic DGFF program

The 2023 results are not confined to planar square lattices. The same fermionic cumulant structure is established for the square and triangular lattices in two dimensions, with the same continuum functional form and lattice-dependent overall constants. For the triangular lattice, the ASM constant is given explicitly by

$\Lambda^g=\Lambda\cup\{g\}$5

The hypercubic lattice results hold for all $\Lambda^g=\Lambda\cup\{g\}$6, and the paper identifies three ingredients for extension to more general embedded graphs: the Matrix–Tree theorem and burning algorithm, a good Green-function approximation to continuum, and isotropic neighborhoods (Chiarini et al., 2023).

By contrast, the fully developed logCFT identification in the 2025 work is specific to two dimensions, planar square-lattice Jordan domains, Dirichlet boundary conditions, simply connected continuum domains, and local fields built from finitely many nearby Grassmann variables modulo null fields (Adame-Carrillo et al., 5 Aug 2025). The paper proves convergence of local-field correlation functions, not a full path-space continuum fermionic field theory.

A useful comparison is the bosonic program for the discrete GFF. The 2024 bosonic work organizes local fields as a quotient by null fields, equips them with Heisenberg and Virasoro actions, and shows that renormalization is governed by the eigenvalue of $\Lambda^g=\Lambda\cup\{g\}$7 (Adame-Carrillo et al., 2024). This suggests a structural parallel: the fDGFF realizes the same general local-field/CFT strategy in a fermionic, logarithmic setting, with symplectic-fermion anticommutation relations and $\Lambda^g=\Lambda\cup\{g\}$8 replacing the bosonic Heisenberg picture and $\Lambda^g=\Lambda\cup\{g\}$9.

In this sense, the fDGFF is both a discrete fermionic Gaussian theory and a rigorous bridge between lattice combinatorics and logarithmic conformal field theory. On finite graphs it yields exact determinant and cumulant formulas for UST and ASM observables; in two dimensions it supports a complete local-field theory whose scaling limit is the symplectic fermions logCFT (Chiarini et al., 2023, Adame-Carrillo et al., 5 Aug 2025).