Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

Abstract

We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a tileable surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the partition function of cycle-rooted spanning forests weighted by the monodromy of the unitary connection on the vector bundle, to the corresponding zeta-regularized determinants. As a consequence, we establish open problems 2 and 4, formulated by Kenyon in 2000. The spectral theory on discretizations of flat surfaces, Fourier analysis on discrete square and the analytic methods used in the proof of Ray–Singer conjecture lie in the core of our approach.

Appendix: heat kernel and analytic torsion

The main goal of the appendix is to recall some folklore results about the heat kernel and the analytic torsion. More precisely, in Appendix A.1, we recall the asymptotics of the heat kernel on the surfaces with conical singularities. In Appendix A.2, we recall the Kronecker limit formula and its relation with the calculation of the analytic torsion of tori and square. We also recall the asymptotic expansion of the determinant of the discrete Laplacian on mesh graphs.

1.1 Small-time asymptotic expansion of the heat kernel

In this section we will prove Proposition 2.5. The proof is done by reducing the small-time asymptotic expansion of the heat kernel on the general flat surface to a number of model cases, for which the explicit calculations are possible.

First, let’s prove that for any $l \in {\mathbb {N}}$, $\epsilon > 0$ there are $c, C > 0$ such that for any $x \in \Sigma$ satisfying ${\mathrm{dist}}_{\Sigma }(x, {\mathrm{Con}}(\Sigma ) \cup {\mathrm{Ang}}(\Sigma )) > \epsilon$, and any $0< t < 1$, we have

$$\begin{aligned} \Bigg \Vert \nabla ^l_{x} \exp (-t \Delta _{\Sigma }^{F})(x, \cdot ) \Bigg \Vert _{L^2(\Sigma {\setminus } B_{\Sigma }(x, \epsilon / 2) )} \le C\exp \Bigg (- \frac{c}{t} \Bigg ). \end{aligned}$$

(A.1)

The estimate (A.1) can be proven by a variety of different methods. We do it by using finite propagation speed of solutions of hyperbolic equations and interior elliptic estimates.

More precisely, for $r > 0$, we introduce smooth even functions (cf. [46,  (4.2.11)])

$$\begin{aligned} \begin{aligned}&K_{t, r}(a) := \int _{- \infty }^{+ \infty } \exp (\sqrt{-1}v \sqrt{2t} a) \exp \Bigg ( -\frac{v^2}{2} \Bigg ) \Bigg (1 - \psi \Bigg ( \frac{\sqrt{2t} v}{r} \Bigg )\Bigg ) \frac{d v}{\sqrt{2 \pi }}, \\&G_{t, r}(a) := \int _{- \infty }^{+ \infty } \exp (\sqrt{-1}v \sqrt{2t} a) \exp \Bigg ( -\frac{v^2}{2} \Bigg ) \psi \Bigg ( \frac{\sqrt{2t} v}{r} \Bigg ) \frac{d v}{\sqrt{2 \pi }}, \end{aligned} \end{aligned}$$

(A.2)

where $\psi : {\mathbb {R}}\rightarrow [0, 1]$ is a cut-off function satisfying

$$\begin{aligned} \psi (u) = {\left\{ \begin{array}{ll} 1 &{} \text { for } |u| < 1/2, \\ 0 &{} \text { for } |u| > 1. \\ \end{array}\right. } \end{aligned}$$

(A.3)

Let ${\widetilde{K}}_{t,r}, {\widetilde{G}}_{t,r} : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}$ be the smooth functions given by ${\widetilde{K}}_{t,r}(a^2) = K_{t,r}(a), {\widetilde{G}}_{t,r}(a^2) = G_{t, r}(a)$. Then the following identity holds

$$\begin{aligned} \exp (-t \Delta _{\Sigma }^{F} ) = {\widetilde{G}}_{t,r}(\Delta _{\Sigma }^{F}) + {\widetilde{K}}_{t,r}(\Delta _{\Sigma }^{F}). \end{aligned}$$

(A.4)

By the finite propagation speed of solutions of hyperbolic equations (cf. [46,   Theorems D.2.1, 4.2.8]), the section ${\widetilde{G}}_{t,r}(\Delta _{\Sigma }^{F}) \big (z, \cdot \big )$, $z \in \Sigma$, depends only on the restriction of $\Delta _{\Sigma }^{F}$ onto the ball $B_{\Sigma }(z, r)$ of radius r around z. Moreover, we have

$$\begin{aligned} {\mathrm{supp}}\, {\widetilde{G}}_{t,r}(\Delta _{\Sigma }^{F}) \big (z, \cdot ) \subset B_{\Sigma }(z, r). \end{aligned}$$

(A.5)

Remark that in [46,   Theorems D.2.1, 4.2.8], authors consider smooth manifolds, but as their reasoning essentially relies on the energy estimate, the proof of which is local and depends only on the validity of the Green’s identities, which hold in our setting according to [29,  Proposition 2.4], it will hold in our setting as well. From (A.4) and (A.5), we get

$$\begin{aligned} \exp (-t \Delta _{\Sigma }^{F})(z, y) = {\widetilde{K}}_{t,r}(\Delta _{\Sigma }^{F})(z, y) \quad \text {if} \quad {\mathrm{dist}}(z, y) > r. \end{aligned}$$

(A.6)

From (A.2), for any $r_0 > 0$ fixed, there exists $c' > 0$ such that for any $m \in {\mathbb {N}}$, there is $C>0$ such that for any $t \in ]0, 1], r > r_0, a\in {\mathbb {R}}$, the following inequality holds (cf. [46,  (4.2.12)])

$$\begin{aligned} |a|^m | K_{t, r}(a) | \le C \exp (- c' r^2/t ). \end{aligned}$$

(A.7)

Thus, by (A.7), for $t \in ]0,1], r > r_0, a \in {\mathbb {R}}_+$, we have

$$\begin{aligned} |a|^m | {\widetilde{K}}_{t, r}(a) | \le C \exp ( -c' r^2/ t ). \end{aligned}$$

(A.8)

Now, by (A.8), there exists $c' > 0$ such that for any $k \in {\mathbb {N}}$, there is $C > 0$ such that for any $t \in ]0,1]$ and $r > r_0$, we have

$$\begin{aligned} \Vert (\Delta _{\Sigma }^{F})^k {\widetilde{K}}_{t, r}( \Delta _{\Sigma }^{F} ) \Vert ^{0}_{L^2(\Sigma )} \le C \exp ( - c' r^2/ t), \end{aligned}$$

(A.9)

where $\Vert \cdot \Vert ^{0}_{L^2(\Sigma )}$ is the operator norm between the corresponding $L^2$-spaces. By interior elliptic estimates, applied in a $\epsilon /2$-neighborhood of x, we deduce that for any $l \in {\mathbb {N}}$, there is $C' > 0$, such that for any $t \in ]0,1]$ and $r > r_0$, we have

$$\begin{aligned} \Bigg \Vert \nabla ^l_{x} {\widetilde{K}}_{t, r}( \Delta _{\Sigma }^{F} ) (x, \cdot ) \Bigg \Vert _{L^2(\Sigma )} \le C' \exp ( - c' r^2/ t). \end{aligned}$$

(A.10)

We get (A.1) from (A.6) and (A.10) by taking $r = \epsilon / 2$.

Now, by using (A.1), we compare the small-time expansions of the heat kernels on $\Sigma$ and on some model manifolds. To do so, we prove Duhamel’s formula, which, to simplify the presentation, we formulate in a vicinity of a conical point.

We fix $P \in {\mathrm{Con}}(\Sigma )$. We denote $\alpha = \angle (P)$ and consider the infinite cone $C_{\alpha }$, (2.2), with the induced metric (2.1). We denote by $\Delta _{C_{\alpha }}$ the Friedrichs extension of the Riemannian Laplacian on $C_{\alpha }$. Let $\epsilon > 0$ be such that $B_{\Sigma }(P, \epsilon )$ is isometric to $C_{\alpha , \epsilon } := B_{C_{\epsilon }}(0, \epsilon )$. From now on, we identify those neighborhoods implicitly.

For $x, y \in C_{\alpha }$ and $t > 0$, we define

$$\begin{aligned} \begin{aligned}&F(x, y, t) := \exp (- t \Delta _{\Sigma }^{F})(x, y) - \exp (- t \Delta _{C_{\alpha }})(x, y), \\&G(x, y, s) := \int _{C_{\alpha , \epsilon }} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z) \cdot \exp (- s \Delta _{C_{\alpha }})(z, y) dv_{C_{\epsilon }}(z). \end{aligned} \end{aligned}$$

(A.11)

Then by the definition of the heat kernel, we have

$$\begin{aligned} \begin{aligned}&\lim _{s \rightarrow t-} G(x, y, s) = \exp (- t \Delta _{C_{\alpha }})(x, y), \\&\lim _{s \rightarrow 0+} G(x, y, s) = \exp (- t \Delta _{\Sigma }^{F})(x, y). \end{aligned} \end{aligned}$$

(A.12)

From (A.12), we deduce that

$$\begin{aligned} \exp (- t \Delta _{C_{\alpha }})(x, y) - \exp (- t \Delta _{\Sigma }^{F})(x, y) = \int _{0}^{t} \frac{d G(x, y, s)}{ds} ds. \end{aligned}$$

(A.13)

However, by the definition of the heat kernel, we have

$$\begin{aligned} \frac{d G(x, y, s)}{ds}= & {} \int _{C_{\alpha , \epsilon }} \Bigg ( \Delta _{\Sigma , x} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z) \Bigg ) \cdot \exp (- s \Delta _{C_{\alpha }})(z, y) dv_{C_{\alpha , \epsilon }}(z)\nonumber \\&- \int _{C_{\alpha , \epsilon }} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z) \cdot \Bigg ( \Delta _{C_{\alpha }, z} \exp (- s \Delta _{C_{\alpha }})(z, y) \Bigg ) dv_{C_{\alpha , \epsilon }}(z),\nonumber \\ \end{aligned}$$

(A.14)

where by $\Delta _{\Sigma , x}$ and $\Delta _{C_{\alpha }, z}$ we mean the Laplace operators acting on variables x and z respectively. By the symmetry of the heat kernel, we have

$$\begin{aligned} \Delta _{\Sigma , x} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z) = \Delta _{\Sigma , z} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z). \end{aligned}$$

(A.15)

Now, since both operators $\Delta _{\Sigma }^{F}$, $\Delta _{C_{\alpha }}$ come from Friedrichs extension of the Riemannian Laplacian, for x and y fixed, the functions $\exp (- (t-s) \Delta _{\Sigma }^{F})(x, \cdot )$ and $\exp (- s \Delta _{C_{\alpha }})(\cdot , y)$ are in the domain of Friedrichs extension. By Green’s identity, cf. [29,  Proposition 2.4], applied to $C_{\alpha , \epsilon }$, and (A.13), (A.14), (A.15), we deduce

$$\begin{aligned}&\exp (- t \Delta _{C_{\alpha }})(x, y) - \exp (- t \Delta _{\Sigma }^{F})(x, y)\nonumber \\&\quad = \int _{0}^{t} \int _{\partial C_{\alpha , \epsilon }} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z) \cdot \Bigg ( \frac{\partial }{\partial n_z} \exp (- s \Delta _{C_{\alpha }})(z, y)\Bigg ) dv_{\partial C_{\alpha }}(z) ds\nonumber \\&\qquad - \int _{0}^{t} \int _{\partial C_{\alpha , \epsilon }} \Bigg ( \frac{\partial }{\partial n_z} \exp (- (t-s) \Delta _{\Sigma }^{F})(x, z) \Bigg ) \cdot \exp (- s \Delta _{C_{\alpha }})(z, y) dv_{\partial C_{\alpha }}(z) ds .\nonumber \\ \end{aligned}$$

(A.16)

The formulas of type (A.16) are also known as Duhamel’s formulas (cf. [4,   Theorem 2.48]).

From (A.1), (A.16), applied for $y = x$ and integrated over $\epsilon$ from $\epsilon _0/2$ to $\epsilon _0$, and Cauchy inequality, we deduce that there are $c, C > 0$ such that for any $0< t < 1$, we have

$$\begin{aligned} \Bigg | \int _{C_{\alpha , \epsilon /2}} \Bigg ( \exp (- t \Delta _{C_{\alpha }})(x, x) - \exp (- t \Delta _{\Sigma }^{F})(x, x) \Bigg ) dv_{C_{\alpha , \epsilon /2}}(x) \Bigg | \le C \exp \Bigg ( - \frac{c}{t} \Bigg ).\nonumber \\ \end{aligned}$$

(A.17)

Similarly, for $Q \in {\mathrm{Ang}}(\Sigma )$, we denote $\beta = \angle (Q)$ and consider the infinite angle $A_{\beta }$ with the induced metric (2.1). We denote by $\Delta _{A_{\beta }}$ the Friedrichs extension of the Riemannian Laplacian with Neumann boundary conditions on $\partial A_{\beta }$. We fix $\epsilon > 0$ in such a way that $B_{\Sigma }(\epsilon , Q)$ is isometric to $B_{A_{\beta }}(\epsilon , 0)$. Similarly to (A.17), we deduce that there are $c, C > 0$ such that for any $0< t < 1$:

$$\begin{aligned} \Bigg | \int _{C_{\alpha , \epsilon /2}} \Bigg ( \exp (- t \Delta _{A_{\beta }})(x, x) - \exp (- t \Delta _{\Sigma }^{F})(x, x) \Bigg ) dv_{A_{\beta , \epsilon /2}}(x) \Bigg | \le C \exp \Bigg ( - \frac{c}{t} \Bigg ).\nonumber \\ \end{aligned}$$

(A.18)

Now, let $R \in \partial \Sigma$ satisfy ${\mathrm{dist}}_{\Sigma }(R, {\mathrm{Con}}(\Sigma ) \cup {\mathrm{Ang}}(\Sigma )) > \epsilon$. We consider a half plane ${\mathbb {H}}= \{(x, y) \in {\mathbb {R}}^2 : y \ge 0 \}$ and identify $0 \in {\mathbb {H}}$ with R. Then $B_{\Sigma }(\epsilon , R)$ is isometric to $B_{{\mathbb {H}}}(\epsilon , 0)$. We denote by $\Delta _{{\mathbb {H}}}$ the self-adjoint extension of the standard Laplacian with Neumann boundary conditions on $\partial {\mathbb {H}}$. Similarly to (A.17), we deduce that there are $c, C > 0$ such that for any $0< t < 1$, we have

$$\begin{aligned} \Bigg | \int _{B_{{\mathbb {H}}}(0, \epsilon /2)} \Bigg ( \exp (- t \Delta _{{\mathbb {H}}})(x, x) - \exp (- t \Delta _{\Sigma }^{F})(x, x) \Bigg ) dv_{{\mathbb {H}}}(x) \Bigg | \le C \exp \Bigg ( - \frac{c}{t} \Bigg ).\nonumber \\ \end{aligned}$$

(A.19)

Finally, let $R \in \Sigma$ satisfy ${\mathrm{dist}}_{\Sigma }(R, {\mathrm{Con}}(\Sigma ) \cup \partial \Sigma ) > \epsilon$. We consider the real plane ${\mathbb {R}}^2$ and identify $0 \in {\mathbb {R}}^2$ with R. Then $B_{\Sigma }(\epsilon , R)$ is isometric to $B_{{\mathbb {R}}^2}(\epsilon , 0)$. We denote by $\Delta _{{\mathbb {R}}^2}$ the self-adjoint extension of the standard Laplacian. Similarly to (A.17), we deduce that there are $c, C > 0$ such that for any $0< t < 1$, we have

$$\begin{aligned} \Bigg | \int _{B_{{\mathbb {R}}^2}(0, \epsilon /2)} \Bigg ( \exp (- t \Delta _{{\mathbb {R}}^2})(x, x) - \exp (- t \Delta _{\Sigma }^{F})(x, x) \Bigg ) dv_{{\mathbb {R}}^2}(x) \Bigg | \le C \exp \Bigg ( - \frac{c}{t} \Bigg ).\nonumber \\ \end{aligned}$$

(A.20)

To conclude, the estimates (A.17) - (A.20) show that to study the asymptotic expansion of ${{\mathrm{Tr}}} \big [ \exp (- t \Delta _{\Sigma }^{F}) \big ]$, as $t \rightarrow 0$, it is enough to consider the analogous problem for a number of model cases: ${\mathbb {R}}^2, {\mathbb {H}}, C_{\alpha } \text { for } \alpha > 0$ and $A_{\beta }$ for $\beta > 0$.

We will start with the simplest one, which is ${\mathbb {R}}^2$. Here, we have the following identity

$$\begin{aligned} \exp (- t \Delta _{{\mathbb {R}}^2})(x, y) = \frac{1}{4 \pi t} \exp \Bigg (- \frac{|x - y|^2}{4t} \Bigg ). \end{aligned}$$

(A.21)

By (A.1), (A.16) and (A.21), we deduce that there are $c, C > 0$ such that for any $x \in \Sigma$ satisfying ${\mathrm{dist}}_{\Sigma }(x, {\mathrm{Con}}(\Sigma ) \cup \partial \Sigma ) > \epsilon$, and any $0< t < 1$, we have

$$\begin{aligned} \Bigg | \exp (- t \Delta _{\Sigma }^{F})(x, x) - \frac{1}{4 \pi t} \Bigg | \le C \exp \Bigg (-\frac{c}{t} \Bigg ). \end{aligned}$$

(A.22)

Now, on the half-plane ${\mathbb {H}}$ and $z_1, z_2 \in {\mathbb {H}}$, we clearly have

$$\begin{aligned} \exp (- t \Delta _{{\mathbb {H}}})(z_1, z_2) = \exp (- t \Delta _{{\mathbb {R}}^2})(z_1, z_2) + \exp (- t \Delta _{{\mathbb {R}}^2})(z_1, \overline{z_2}). \end{aligned}$$

(A.23)

From (A.21) and (A.23), after an easy calculation, we see that for any $k \in {\mathbb {N}}$ and for any $f \in {\mathscr {C}}^{\infty }_{0}({\mathbb {H}})$, there is $C > 0$ such that for $0< t < 1$, we have

$$\begin{aligned}&\Bigg | \int _{{\mathbb {H}}} f(x) \exp (- t \Delta _{{\mathbb {H}}})(x, x) dv_{{\mathbb {H}}}(x) - \frac{1 }{4 \pi t} \int _{{\mathbb {H}}} f(x) dv_{{\mathbb {H}}}(x)\nonumber \\&\quad - \frac{1}{4 \pi \sqrt{t}} \sum _{l = 0}^{2 k + 1} \frac{t^l \Gamma (\frac{l + 1}{2})}{l!} \int _{\partial {\mathbb {H}}}\frac{\partial ^{l} f}{\partial y^{l}}(x, 0) dv_{\partial {\mathbb {H}}}(x) \Bigg | \le C t^{k}. \end{aligned}$$

(A.24)

Let’s now study the cone $C_{\alpha }$, $\alpha > 0$. Carslaw in [14] (cf. also Kokotov [43,   Sect. 3.1]), gave an explicit formula for the heat kernel on $C_{\alpha }$. By using this formula, Kokotov proved in [43,  Proposition 1, Remark 1] that for any $\epsilon > 0$, there are $c, C > 0$ such that for any $0< t < 1$:

$$\begin{aligned} \Bigg | \int _{B_{C_{\alpha }}(0, \epsilon )} \exp (- t \Delta _{C_{\alpha }})(x, x) dv_{C_{\alpha }}(x) - \frac{\alpha \epsilon ^2}{8 \pi t} - \frac{1}{12} \frac{4 \pi ^2 - \alpha ^2}{2 \pi \alpha } \Bigg | \le C \exp \Bigg (-\frac{c}{t} \Bigg ). \end{aligned}$$

(A.25)

Let’s finally study the angle $A_{\beta }$, $\beta > 0$. Recall that in [59,   Theorem 1], Berg-Srisatkunarajah have obtained the small-time asymptotic expansion of the heat trace associated to the Friedrichs extension of the Laplacian on the angle endowed with Dirichlet boundary conditions. In [48,  Proposition 2.1 and (4.2)], Mazzeo-Rowlett used this result, along with (A.25), and an interpretation of a cone through gluing of two angles: with Dirichlet and Neumann boundary conditions, to show that for any and $\epsilon > 0$, there are $c, C > 0$ such that for any $0< t < 1$, we have

$$\begin{aligned} \Bigg | \int _{B_{A_{\beta }}(0, \epsilon )} \exp (- t \Delta _{A_{\beta }})(x, x) dv_{A_{\beta }}(x) - \frac{\beta \epsilon ^2}{8 \pi t} - \frac{2 \epsilon }{4 \sqrt{\pi } \sqrt{t}} - \frac{1}{12} \frac{\pi ^2 - \beta ^2}{2 \pi \beta } \Bigg | \le C \exp \Bigg (-\frac{c}{t} \Bigg ).\nonumber \\ \end{aligned}$$

(A.26)

By (A.17)-(A.20), (A.22), (A.24), (A.25) and (A.26), we easily conclude that Proposition 2.5 holds.

1.2 Kronecker limit formula and the analytic torsion

The goal of this section is to recall some explicit formulas for the analytic torsion and the asymptotic expansion of the determinants of discrete Laplacians on the mesh graphs. The material of this section is classical, and we include it here only to make this paper self-contained.

We start by considering the continuous side of the story. We fix $a, b \in {\mathbb {N}}^*$ and consider a torus ${\mathbb {T}}_{a,b} := {\mathbb {R}}^2 / (a {\mathbb {Z}}\times b {\mathbb {Z}})$. The standard Laplacian $\Delta _{{\mathbb {R}}^2} := - \frac{\partial ^2}{\partial x^2} - \frac{\partial ^2}{\partial y^2}$ descends to Laplacian $\Delta _{{\mathbb {T}}_{a,b}}$, acting on the functions on ${\mathbb {T}}_{a,b}$. Classically, the eigenvalues of $\Delta _{{\mathbb {T}}_{a,b}}$ are given by

$$\begin{aligned} {\mathrm{Spec}}\big ( \Delta _{{\mathbb {T}}_{a,b}} \big ) = \Bigg \{ (2 \pi )^2 \Bigg ( \Bigg ( \frac{n}{a} \Bigg )^2 + \Bigg ( \frac{m}{b} \Bigg )^2 \Bigg ) : n, m \in {\mathbb {Z}}\Bigg \}. \end{aligned}$$

(A.27)

So the associated spectral zeta function $\zeta _{{\mathbb {T}}_{a,b}}(s)$ is given by

$$\begin{aligned} \zeta _{{\mathbb {T}}_{a,b}}(s) = (2 \pi )^{-2s} \sum _{(m, n) \ne (0, 0)} \frac{1}{((n/a)^2 + (m/b)^2)^s}. \end{aligned}$$

(A.28)

We rewrite it in the following form

$$\begin{aligned} \zeta _{{\mathbb {T}}_{a,b}}(s) = (ab)^s E(z, s), \end{aligned}$$

(A.29)

where $z := \sqrt{-1}\frac{a}{b}$, and

$$\begin{aligned} E(z, s) = (2 \pi )^{-2s} \sum _{(m, n) \ne (0, 0)} \frac{{\text {Im}}(z)^s}{|nz + m|^{2s}}. \end{aligned}$$

(A.30)

The function E(z, s) is the Eisenstein series multiplied by $(2 \pi )^{-2s}$ and thus it admits a meromorphic continuation to ${\mathbb {C}}$. By Kronecker limit formula, it has the following expansion near $s = 0$:

$$\begin{aligned} E(z, s) = - 1 - s \log \Bigg ( \frac{a}{b} \cdot \nu \big ( e^{- \frac{2 \pi a}{b}} \big )^4 \Bigg ) + o(s), \end{aligned}$$

(A.31)

where $\nu : D(1) \rightarrow {\mathbb {C}}$ is the Dedekind eta-function, defined by

$$\begin{aligned} \nu (q) = q^{1/24} \prod _{n \ge 1} (1 - q^n). \end{aligned}$$

(A.32)

By (A.29) and (A.31), we obtain

$$\begin{aligned} \log \det {}' \Delta _{{\mathbb {T}}_{a,b}} = -\zeta _{{\mathbb {T}}_{a,b}}'(0) = \log (ab) + \log \Bigg ( \frac{a}{b} \cdot \nu \big ( e^{- \frac{2 \pi a}{b}} \big )^4 \Bigg ). \end{aligned}$$

(A.33)

Remark that (A.33) goes in line with Duplantier-David [25,  (3.18)] and Corollary 1.3.

Now, consider a square $[0, a] \times [0, b] \subset {\mathbb {C}}$. We consider the Friedrichs extension of the standard Laplacian $\Delta _{[0, a] \times [0, b]}$ endowed with Neumann boundary conditions. The eigenvalues of $\Delta _{[0, a] \times [0, b]}$ are given by

$$\begin{aligned} {\mathrm{Spec}}\big ( \Delta _{[0, a] \times [0, b]} \big ) = \Bigg \{ \pi ^2 \Bigg ( \Bigg ( \frac{n}{a} \Bigg )^2 + \Bigg ( \frac{m}{b} \Bigg )^2 \Bigg ) : n, m \in {\mathbb {N}}\Bigg \}. \end{aligned}$$

(A.34)

So the associated spectral zeta function $\zeta _{[0, a] \times [0, b]}(s)$ is related to the zeta function of torus ${\mathbb {T}}_{a, b}$ by

$$\begin{aligned} \zeta _{[0, a] \times [0, b]}(s) = 4^{s - 1} \zeta _{{\mathbb {T}}_{a, b}}(s) + \frac{1}{2} \Bigg ( \Bigg ( \frac{a}{\pi } \Bigg )^{2s} + \Bigg ( \frac{b}{\pi } \Bigg )^{2s} \Bigg ) \zeta (2s), \end{aligned}$$

(A.35)

where $\zeta (s)$ is the Riemann zeta function. By using identities

$$\begin{aligned} \zeta (0) = -\frac{1}{2}, \qquad \zeta '(0) = - \frac{\log (2\pi )}{2}, \end{aligned}$$

(A.36)

and (A.31), (A.33), we get the following expression

$$\begin{aligned} \log \det {}' \Delta _{[0, a] \times [0, b]} = -\zeta _{[0, a] \times [0, b]}'(0) = \frac{3}{4} \log (ab) + \frac{1}{4} \log \Bigg ( \frac{a}{b} \cdot \nu \big ( e^{- \frac{2 \pi a}{b}}\big )^4 \Bigg ) + \frac{3}{2} \log (2).\nonumber \\ \end{aligned}$$

(A.37)

We also obtain that $\zeta _{[0, a] \times [0, b]}(0) = -\frac{3}{4}$, which is compatible with (1.12).

Now, let us study the discrete side of the problem. As in Sect. 4.1, construct a family of graphs $A_{an \times bn}$, $n \in {\mathbb {N}}^*$. We use the notation as in Theorem 1.1. Using explicit formula (A.37) Duplantier-David in [25,  (4.7) and (4.23)] proved the following theorem.

Theorem A.1

As $n \rightarrow \infty$, the following asymptotic expansion holds

$$\begin{aligned}&\log (\det {}'\Delta _{A_{an \times bn}}) = \frac{4 G ab}{\pi } \cdot n^2 + \log (\sqrt{2} - 1) \cdot (a + b) \cdot n + \frac{\log (n)}{2}\nonumber \\&\quad + \log ( \det {}' \Delta _{\Sigma }^{F} ) - \frac{\log (2)}{4} + o(1). \end{aligned}$$

(A.38)

We see, in particular that Theorem 1.1 is compatible with Theorem A.1.