A glimpse into the Ultrametric spectrum

TL;DR

The paper investigates whether an ultrametric (p-adic) spectral construction can reproduce the Hardy-Ramanujan entropy scaling of the non-relativistic string. By analyzing tree-graph normal modes under various boundary conditions and then passing to the p-adic unit circle with the Vladimirov derivative, the authors derive an exponentially spaced ultrametric spectrum with exponentially growing degeneracies. A saddle-point evaluation of the corresponding oscillator gas shows that, after averaging over log-periodic fluctuations, the microstate count grows as Ω(E) ∼ exp[ c sqrt{E} ] with a modulating log-periodic factor. This establishes an ultrametric counterpart to two-dimensional thermodynamics and clarifies limitations and connections to Neumann-to-Dirichlet, Moore cages, and potential Lorentzian extensions.

Abstract

The non-relativistic string spectrum is built from integer-spaced energy quanta in such a way that the high-temperature asymptotics, via the Hardy-Ramanujan formula for integer partitions, reduces to standard two-dimensional thermodynamics. Here we explore deformed realizations of this behavior motivated by $p$-adic string theory and Lorentzian versions thereof with a non-trivial spectrum. We study the microstate scaling that results on associating quantum harmonic oscillators to the normal modes of tree-graphs rather than string graphs and observe that Hardy-Ramanujan scaling is not realized. But by computing the eigenvalues of the derivative operator on the $p$-adic circle and by determining the eigenspectrum of the Neumann-to-Dirichlet operator, we uncover a spectrum of exponentially growing energies but with exponentially growing degeneracies balanced in such a way that Hardy-Ramanujan scaling is realized, but modulated with log-periodic fluctuations.

Figures (8)

• Figure 1: The fractal tree for $p=2$ with a cutoff imposed at $L=4$ edges from a central vertex.
• Figure 2: Plot of the values of $\lambda$ grouped according to their associated degeneracies for the finite fractal tree with $p=7$ and $L=14$. At the lowest degeneracy, the smallest value of $\lambda$ is exactly zero, while at higher degeneracies the smallest values are slightly above zero.
• Figure 3: Plot of the values of $\lambda$ for the 54 solutions to the equation $h^{53}(1-\lambda)=0$ for $p=7$. The value of the first solution is about $1.191\cdot 10^{-45}$.
• Figure 4: By identifying boundary vertices of a finite tree, one can construct a vertex-transitive graph. For this example with $p=2$ and $L=3$, the resulting graph $\mathcal{T}^{(c)}_{2,3}$ is the Heawood graph.
• Figure 5: Computation of the eigenvalue for the multiplicative character of conductor two for $p=3$. The 3-adic units $\mathbb{U}_3$ are congruent to 1, 4, 7, 2, 5, and 8 mod 9. These six numbers form a group under multiplication, and we can pick two as a generator $g=2$. The eigenvalue has a four-fold degeneracy as $j$ can assume any of the values 1, 2, 4, and 5.