# Regularized Laplacian determinants of self-similar fractals

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## Abstract

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.

## Keywords

Regularized determinant Fractal Laplacian Spectral zeta functions Sierpiński gasket## Mathematics Subject Classification

28A80 05C30 58C40 58J52 81Q10## 1 Introduction

The spectrum of the Laplace operator on fractals has been the focus of considerable work, see e.g., [9, 10, 19, 22, 29, 30, 32, 33, 35]. Given a post-critically finite (p.c.f.) self-similar set (see [21] for the definition of p.c.f.), one can compute its spectral dimension \(d_\mathrm{s}\) and walk dimensions \(d_\mathrm{w}\), and these dimensions are connected via the Einstein relation \(d_\mathrm{s}=2\frac{d_\mathrm{f}}{d_\mathrm{w}}\) where \(d_\mathrm{f}\) is the Hausdorff dimension. In [12, 31, 37, 38], the spectral zeta functions have been studied, and while they are defined initially only for \(s>\frac{d_\mathrm{s}}{2}\), they are shown to meromorphically extend to the entire complex plane. Their poles, also called *complex dimensions* [24], are studied in [31] and it is proven that for a large class of p.c.f. fractals with symmetries, that the poles can only be on the imaginary axis or on the axis where \({Re}(s)=\frac{d_\mathrm{s}}{2}\).

Our long-term motivation comes from quantum physics, in particular such recent papers as [1, 2, 3, 4, 5, 14, 15, 25, 28, 36] and more classical works [16, 17, 20, 23]. Our immediate mathematical motivation is twofold. On the one hand, it comes from the following statement found in [12] and [13]:

“If there were no poles on the imaginary axis, then \(e^{-\zeta '_{\varDelta }(0)}\) would be the regularized product of eigenvalues or the Fredholm determinant of \(\varDelta \).”

*d*-tuple of positive integers parametrized by \(u \in {\mathbb {Z}}\), such that for each

*j*, we have \(\frac{n_j(u)}{u} \rightarrow a_j\) as \(u \rightarrow \infty \). One then defines the

*d*-dimensional discrete torus as the product space

*A*is the diagonal matrix with entries \(a_j\) and \(V(a)=a_1\cdots a_d\), the authors of [11] established the formula

*RT*,

*A*is the real torus \(A {\mathbb {Z}}^d / {\mathbb {R}}^d\), and \({\mathcal {I}}\) is a specific special function. A variation of this result was also studied in [40].

*f*as \(\widetilde{\lim _{x \rightarrow \infty }} f(x)=a_{00}.\) Then as in [40] we can restate the above result as

*pq*-model on the unit interval. All three examples satisfy spectral decimation, which leads to closed-form expressions for the spectral zeta functions and the Laplacian determinants. However, only for the double Sierpiński gaskets and the double

*pq*-model do we have exact analogs of (1.1). Details will be described in subsequent sections, after a review of basic notions from analysis on fractals and graph theory.

## 2 Notions of analysis on fractals and graph theory

*X*,

*d*), and injective contractions \(F_i: X \rightarrow X\), \(i \in \{1,2,\ldots , m\}\), there exists a unique non-empty compact subset

*K*of

*X*that satisfies

*K*is a sequence of approximating graphs \(\{G_n : n\ge 0\}\), defined as follows. Let \(V_0\) be the set consisting of the essential fixed points of the maps \(F_i\) and \(G_{0}\) be the complete graph on \(V_{0}\). For \(n\ge 1\), we define inductively

*K*. Given the energy, we can define the Laplacian which is the main focus of our study. For \(u \in \hbox {dom}{\mathcal {E}}\) we say \(u \in \hbox {dom}\varDelta _{\mu }\) and \(\varDelta _{\mu }u=f\) if

### 2.1 Spectral decimation and zeta functions

*A*is a finite set and

*R*is a rational function. For the limit to exist the elements of the preimages \(R^{-(n)}(w)\) must be chosen appropriately. The value

*m*stands for the so-called generation of birth of the eigenvalue

*w*and is independent of

*n*; more information may be found at [12, 19, 34, 38]. In many cases, this rational function turns out to be a polynomial. The quantity \(\lambda \) is also known as the time scaling factor.

### 2.2 Counting spanning trees in fractal graphs

*combinatorial graph Laplacian*of a graph \(G_n\) as \(\varDelta _n=D-A\), and the

*probabilistic graph Laplacian*as \({\mathcal {L}}_n=I-D^{-1}A\), where

*D*is the diagonal degree matrix and

*A*is the adjacency matrix. Then the number of spanning trees in \(G_n\) may be expressed in either of two ways:

*asymptotic complexity constant*of \((G_n)\), studied in [26],

*A*and

*B*. If the rational function associated with spectral decimation is of the form \(R(z)=\frac{P(z)}{Q(z)}\) with degree

*d*, and \(P_d\) is the leading coefficient of

*P*, then

## 3 Zeta function of the diamond fractal

### Proposition 3.1

### Proof

### Remark 3.1

At first glance, the complex dimensions would be located at the positions such that \(4^s=4\) which are \(s=1+\frac{ik\pi }{\log {2}}\) and at \({Re}(s)=\frac{\log {2}}{\log {4}}=\frac{1}{2}\) due to the poles of the polynomial zeta functions. However, it was shown in [31] that the complex dimensions can only be on the imaginary axis, which we have proven is not the case, and at \({Re}(s)=\frac{d_\mathrm{s}}{2}=1\) and thus we can deduce that all the poles of the polynomial zeta functions must be canceled by the zeros of the geometric part which we observe that is indeed the case for \({Re}(s)=\frac{1}{2}\).

### Remark 3.2

The value \(\log {2}\) can also be interpreted as the tree entropy or the asymptotic complexity constant of the sequence of the fractal graphs approximating the diamond fractal. Thus \(\log {\det {\mathcal {L}}}=-\frac{10}{9}c\).

## 4 Zeta function of double Sierpiński gaskets

*N*identifications. Then it becomes a fractal without boundary and the spectrum of the Laplace operator is the union of the Dirichlet and Neumann spectra with added multiplicities.

### Proposition 4.1

### Proof

### Remark 4.1

As in the case of the diamond fractal, the zeros of the geometric part cancel all the poles of the polynomial zeta functions, and the only poles that remain are at \({Re}(s)=\frac{d_s}{2}=\frac{\log {N}}{\log {(N+2)}}\).

We establish now a result analogous to [11].

### Corollary 4.1

*c*is the asymptotic complexity constant which is

### Proof

### Remark 4.2

### Remark 4.3

By using Kirchhoff’s Matrix-Tree theorem and the above calculations, we can also calculate the number of spanning trees for the single \(N-1\)-dimensional Sierpiński gasket confirming the formula conjectured in [8] and first proven via a different methodology in [39]. The asymptotic complexity constant for the single and double pre-fractal Sierpiński graphs are the same.

## 5 Zeta function of the double pq-model on the unit interval

Spectral decimation has been carried in [38] for the Neumann case with rational function \(R_p(z)=\frac{1}{pq}z(\frac{z^2}{4}+\frac{3z}{2}+2+pq)\) and the following is obtained.

### Proposition 5.1

We can easily see that \(R^{-1}(0)=\{0,-2-2p,-2-2q\}\) and \(R^{-1}(-4)=\{-4,-2+2p,-2+2q\}\) and thus the spectrum is obtained as follows

### Proposition 5.2

*pq*-model is given by

Before we give this proof, we must calculate the Dirichlet spectrum for the single *pq*-model. By solving the Dirichlet eigenvalue equation on the first level, we see that the eigenvalues are \(-1-p\) and \(-1+p\). These eigenvalues are initial and they show up at every level, and we encounter no exceptional eigenvalues by taking their preiterates. This means that the spectrum is of the following form.

Thus while for the Neumann spectrum we have that \(\dim \varDelta _n=3\dim \varDelta _{n-1}-2\) for the Dirichlet spectrum we have that \(\dim \varDelta _n=3\dim \varDelta _{n-1}+2\). Then the proof of the proposition goes as follows.

### Proof

### Remark 5.1

The location of the poles must necessarily coincide with the location of the poles of the polynomial zeta functions and are thus at \({Re}(s)=\frac{\log {3}}{\log {\lambda }}\).

Then as in [11] we establish that the logarithm of the regularized determinant appears as a constant in the logarithm of the determinant of the discrete graph Laplacians.

### Corollary 5.1

*pq*-model, we have that

### Proof

## Notes

### Acknowledgements

We thank Professors Robert S. Strichartz, Gerald Dunne and Peter Grabner for helpful discussions and Anders Karlsson for suggesting the problem. The last-named author would also like to thank the Mathematics Department at the University of Connecticut for the hospitality during his research stay. Research of the first named author is supported by the Simons Foundation (via a Collaboration Grant for Mathematicians #523544). Research of the second-named author is supported in part by NSF Grant DMS-1613025.

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