See also Ref. [44] for recent work on quantum hierarchical models and Ref. [45] for recent work on critical phenomena on the Bethe lattice.

See also Refs. [47, 48, 49, 50] for alternative approaches to $p$-adic quantum mechanics and for related notions of statistical field theory [51, 52].

See e.g. Eq. (10) of Ref. [57].

See e.g. Eq. (3) of Ref. [59].

A complication present more generally is that a clean separation of oscillations along independent and identical transverse directions does not subsist beyond the special case of chain graphs. In the more general case, the form of the oscillation modes that do not change edge lengths to first order in displacement will depend on the details of the graph’s embedding into target space, as studied within the field of rigidity theory.

While the equation $h^{\ell -1}(1-\lambda )=0$ can be shown to imply Eq. (36), the converse is not true as Eq. (36) has two additional roots. However, as the smallest root is shared by the two equations, the approximation (37) remains valid.

In the simplest model where we consider the fields ${\varphi }_x$ as representing small transverse oscillations of a cobweb stretched along a single spatial direction and think of the vertices as beads of mass $m$, with the edges connecting them serving as springs with a uniform spring constant $k$, we have the relation $\lambda =m{\omega }^2/k$. To obtain a finite total mass $M=m{N}_{p,L}$, the individual bead mass $m$ must scale as $1/N_{p,L}$. Using the fact that the spring constants of springs in parallel are directly additive while those of springs in series are reciprocally additive, one finds that $K=(2·\frac{2}{p+1}{\sum }_{n=0}^{L-1}{p}^{-n}{)}^{-1}k=\frac{p^2-1}{4p(1-p^{-L})}k$. For $p>1$, $K$ has a finite limit as $L\to \infty$ whereas in the case $p=1$ the relation is $K=k/(2L)$. For $p=1$ then, using Eqs. (34) and (35), the $L\to \infty$ limit at fixed $M$ and $K$ then reproduces the familiar string spectrum with integer spacing: $\omega =\sqrt{\frac{k\lambda }{m}}=\sqrt{\frac{K2L(2L+1)\lambda }{M}}\underset{L\to \infty }{\to }\pi \sqrt{\frac{K}{M}}n$ with $n\in {\mathbb{N}}_0$. For $p>1$ meanwhile, $K$ can be held fixed in the infinite system limit, with the normal mode spectrum determined from the infinite $L$ limit of ${\omega }^2=\frac{\lambda k}{m}=(\lambda N_{p,L})\frac{4p(1-p^{-L})}{p^2-1}\frac{K}{M}$, which results in Eq. (38) with ${\omega }_0=2\sqrt{pK/M}$.

On a technical note, the roots for different values of $\ell$ sometimes overlap so that one must add together the respective multiplicities to get the total degeneracy. E.g., for $p=2$ and $L=6$, the eigenvalue $\lambda =3$ has a total degeneracy of 64, which comes from a multiplicity-1 ${\lambda }_{L,n}$ eigenvalue, a multiplicity-3 ${\lambda }_{L-2,n}$ eigenvalue, a multiplicity-12 ${\lambda }_{L-4,n}$ eigenvalue, and a multiplicity-48 ${\lambda }_{L-6,n}$ eigenvalue.

One could attempt to obtain a nontrivial spectrum through the introduction by hand of a mass term that effectively replaces the graph Laplacian $\square$ in Eq. (21) with $\square -(p+1-2\sqrt{p})$. But this does not lead to a sensible spectrum either. For since ${\lambda }_{L-\ell ,n}-(p+1-\sqrt{p})=\sqrt{p}\frac{n^2{\pi }^2}{L^2}+\mathcal{O}(1/L^4)$ with no $\ell$ dependence on the rhs, this mass shift results in an infinitely degenerate ground state on taking the $L\to \infty$ limit.

The whole analysis can also be straightforwardly extended to the cases when $p$ is a prime power: $p={\mathfrak{p}}^n$ with $n\in \mathbb{N}$ and $\mathfrak{p}\in \mathbb{P}$. In this case the set of $p$-adic numbers ${\mathbb{Q}}_p$ is replaced with the degree $n$ unramified extension of ${\mathbb{Q}}_p$.

On a technical note, since the boundary of the tree also includes the point at infinite, the boundary space should more precisely be identified with the projective space ${\mathbb{P}}^1({\mathbb{Q}}_p)$.

While the present work focuses on the entropy and free energy of nonrelativistic quantum systems and everything is real, in the context of relativistic CFT correlators, such log-periodic fluctuations would be indicative of a complex CFT [84, 85].

The founding references are Refs. [37, 38], see also Ref. [92] for its tensor-network implementation and Refs. [93, 94, 95, 96, 97] for more recent work on the subject.

An analytic closed-form expression for the one-loop cosmological constant of the bosonic string was written down only last year [103].