A glimpse into the Ultrametric spectrum

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.

version 2: separate treatment of $p=2$, typos fixed, extra discussion, more references. This version matches the journal version

References in corpus (46)

• Walking, Weak first-order transitions, and Complex CFTs
• Programmable Interactions and Emergent Geometry in an Atomic Array
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$
• Where is String Theory?
• Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude
• $p$-adic AdS/CFT
• Treelike interactions and fast scrambling with cold atoms
• Thermal Duality and Hagedorn Transition from p-adic Strings
• Carving out the Space of Open-String S-matrix
• Linear Regge trajectories from worldsheet lattice parton field theory
• Veneziano Variations: How Unique are String Amplitudes?
• Generalized Veneziano and Virasoro amplitudes
• Stringy Dynamics from an Amplitudes Bootstrap
• On the Quantum Density of States and Partitioning an Integer
• Complex dimensions and their observability
• On the unitarity and low energy expansion of the Coon amplitude
• Multiparticle Factorization and the Rigidity of String Theory
• Accumulation-Point Amplitudes in String Theory
• Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon
• On Fine Structure of Strings: The Universal Correction to the Veneziano Amplitude
• A Meinardus theorem with multiple singularities
• Spin in $p$-adic AdS/CFT
• Bootstrap Principle for the Spectrum and Scattering of Strings
• Continuum limits of sparse coupling patterns
• Cutting the Coon Amplitude
• On the positivity of Coon amplitude in D=4
• Closed Strings and Weak Gravity from Higher-Spin Causality
• p-Adic Statistical Field Theory and Deep Belief Networks
• Uniqueness Criteria for the Virasoro-Shapiro Amplitude
• p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines
• Oscillatory critical amplitudes in hierarchical models and the tail of the Harris random variable
• Positivity of the Hypergeometric Coon Amplitude
• Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices
• Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model
• Complete Solution of the Tight Binding Model on a Cayley Tree: Strongly Localised versus Extended States
• Coon unitarity via partial waves or: how I learned to stop worrying and love the harmonic numbers
• Statistical Mechanics Approach to the Holographic Renormalization Group: Bethe Lattice Ising Model and p-adic AdS/CFT
• Non-Archimedean Quantum Mechanics via Quantum Groups
• The emergence of Logarithmic-periodic oscillations in Contact process with the topological disorder
• Probing holography in $p$-adic CFT
• Rademacher expansion of modular integrals
• Spreading of a local excitation in a Quantum Hierarchical Model
• Effective field theories on subspaces of the Bruhat-Tits tree
• Finite Temperature at Finite Places
• Toward Holography on Biregular Trees