P-adic numbers and kernels

TL;DR

The paper builds a unified framework connecting $p$-adic numbers, kernel theory, and replica-symmetry breaking (RSB) by representing numbers as hierarchical kernel structures. It extends binary kernels to a canonical $p$-base and a mixed-prime base, deriving ultrametric overlaps and a tree-like kernel description, with the Generalized Random Energy Model (GREM) emerging as a bijective base map on consecutive integers. A kernel formulation of the Primon gas is developed, showing that the canonical spectrum $v_n(p)$ yields $H(n)=\sum_{\ell} v_n(p_{\ell})\log p_{\ell}$ and reproduces the Riemann zeta function via $\zeta(\beta)=\sum_{n\ge1} e^{-\beta H(n)}$, or as a product over primes. The results operationalize arithmetic models through ultrametric kernels, offering a concrete bridge between number theory and kernel-based approaches with potential implications for large deviations in mean-field spin glasses. Overall, the work provides a structured methodology to translate between numerical bases, hierarchical spin representations, and number-theoretic spectra using kernel representations.

Abstract

We discuss the relation between p-adic numbers and kernels in view of a recent large deviation theory for mean-field spin glasses. As an application we show several fundamental properties of numerical bases in kernel language. In particular, we show that the Derrida's Generalized Random Energy Model can be interpreted as a (random) numerical base. We also show an application to the Primon gas and the Riemann Zeta Function by constructing a kernel representation of the Primon gas based on a finite p-base, thereby establishing a concrete link between number theory and kernel theory.

Figures (6)

• Figure 1: Binary kernel describing the first $2^{8}=64$ natural numbers $\mathbb{N}_{64}$ (zero included), the numbers are organized as column of the kernel, ordered from the smaller $0$ on the left to the larger $n=2^{8}-1=63$ to the far right. The index $i$ runs from bottom $1$ to top $N$ (shown on the right). The corresponding base vector to construct the map $M_{2}$ is shown on the left.
• Figure 2: Rescaled kernel, the sum of each column is exactly the original number rescaled with the dyadic norm $2^{-8}=1/64$. The numbers are ordered from smaller $0$ on the left to larger $63$ on the right.
• Figure 3: We can also apply the transformation $N\longrightarrow-N$ that is proposed in ParisiSourl as equivalent for the replica trick. The obtained dual kernel is shown.
• Figure 4: Base kernel $A$. The kernel $A:\left[0,1\right]^{2}\rightarrow\mathbb{R}$ is subdivided into zones organized a according to a tree-indexed partition, $\left(1010\right)_{2}=\left(10\right)_{10}$ is shown. For each index vector $\underline{a}_{i}=a_{1}a_{2}...a_{i}$ there is a corresponding value $A\left(\underline{a}_{i}\right)$. If we take $A\left(\underline{a}_{i}\right)=J_{\underline{a}_{i}}=J_{a_{1}a_{2}...a_{i}}$ with $J_{\underline{a}_{i}}$ independent Gaussians of mean zero and variance $\gamma_{i}^{2}$ the resulting kernel is ultrametric (the scalar product of the columns are ultrametric on average)
• Figure 5: Choosing $A\left(\underline{a}_{i}\right)=2^{N-i}a_{i}$ give the binary numbering back. The pattern in the previous figure is the number $\left(1010\right)_{2}=\left(10\right)_{10}$