Theorem. A convergent expansion over closed walks [fdip-0071]
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  • November 1, 2026

Let the vertices be \(\Lambda =\prod _{j=1}^d\mathbb Z/L_j\mathbb Z\), with \(L_j\ge 3\). Give each edge \((x,j):x\to x+e_j\) weight one and a unitary or orthogonal transport \(U_{x,j}\) on a fixed fiber. For the determinant action, put \(D=2d+m^2\), where \(m>0\), and let \(H_U\) be the covariant adjacency operator. Then \[Q_U=DI-H_U,\qquad \|H_U\|\le 2d<D,\] and \[W(U)=\kappa N\log D- \kappa \sum _{k\ge 1}\frac {\operatorname {tr}_{\mathbb K}(H_U^k)}{kD^k}.\] The series is absolutely convergent, uniformly over all such unitary connections. For \(K\ge 0\), its remainder after length \(K\) satisfies \[|R_K(U)|\le \frac {\kappa N}{K+1} \frac {(2d/D)^{K+1}}{1-2d/D}.\] Each trace is the sum of fiber holonomy traces over all based oriented closed nearest-neighbor walks of length \(k\).

Proof. 

  • November 1, 2026

Define the covariant shift \[(T_j\phi )(x)=U_{x-e_j,j}\phi (x-e_j).\] A permutation of sites and the unitary fiber maps preserve the norm, so \(T_j\) is unitary. Expanding the difference energy gives \[H_U=\sum _{j=1}^d(T_j+T_j^*),\qquad Q_U=(2d+m^2)I-H_U.\] Thus \(H_U\) is self-adjoint and the triangle inequality gives \(\|H_U\|\le 2d\). If its eigenvalues are \(h_1,\ldots ,h_N\), then \(|h_i|/D<1\). Apply the scalar Taylor series for \(\log (1-h_i/D)\) to \(\log \det Q_U=N\log D+\sum _i\log (1-h_i/D)\). There are finitely many eigenvalues and \[|\operatorname {tr}(H_U^k)|\le N(2d)^k,\] so the resulting trace series is absolutely convergent. For \(k\ge K+1\), bound \(1/k\le 1/(K+1)\) and sum the geometric tail. This proves the stated uniform remainder.

The matrix block from \(s(e)\) to \(h(e)\) in \(H_U\) is \(U_e\), and the reverse block is \(U_e^*\). Multiplying \(k\) blocks follows an oriented walk; the diagonal blocks are exactly the closed walks. The ordered product agrees with the transport convention, and taking the fiber trace gives the claimed expansion. Reverse walks pair adjoint products, so the total is real even for a complex fiber.

Mathlib's Real.hasSum_pow_div_log_of_abs_lt_one states, for a real \(x\) with \(|x|<1\), \[\sum _{n\ge 0}\frac {x^{n+1}}{n+1}=-\log (1-x).\] It supplies the scalar series used for each real eigenvalue. Spectral reduction, the walk interpretation, the lattice norm bound and the uniform trace remainder are the additional arguments above; the formal declaration does not state this lattice expansion.

The numerical bound \(2d\) and scalar diagonal \(D\) use this regular lattice and its unit edge weights. For arbitrary weights or degrees one must change the diagonal decomposition and norm estimate; the displayed bound is not asserted unchanged.

Kazakov and Migdal integrate an adjoint matrix field in a different lattice model. Their Section 2, equations (3)–(4), gives a trace-log and loop weights involving \(|\operatorname {tr}U[\Gamma ]|^2\) [kazakov1992induced, p. 2]. This is a comparison with the closed-walk method. The finite real-scalar coefficient below follows from the present proof, not from their large-\(N\) analysis or a continuum identification.