Reflecting Poisson walks and dynamical universality in $p$-adic random matrix theory

TL;DR

The paper defines the reflecting Poisson sea, a bi-infinite Poisson walker system with local reflection, as a universal dynamical limit for the singular-number evolution in products of $p$-adic random matrices. By coupling finite multisets of Poisson walkers and passing to bi-infinite limits, the authors construct $\mathcal{S}^{\mu,2\infty}(T)$ and its edge variant, and prove dynamical convergence for matrix-product processes under minimal $GL_N(\mathbb{Z}_p)$-invariance assumptions. A key innovation is proving dynamical universality without requiring a single-time universality, via Markov-generator asymptotics and robust nonasymptotic linear-algebra bounds, enabling bulk and edge results that mirror classical line ensembles but with local interactions. The work unifies bulk and edge limits for $p$-adic matrix products, showing that wide classes of matrix distributions yield the same dynamic limit, and provides a pathway to translate fixed-time limits into multi-time dynamical statements. This advances understanding of cokernel growth and singular-number dynamics in non-Archimedean random matrices, with potential links to random abelian $p$-groups and noncommutative growth processes.

Abstract

We prove dynamical local limits for the singular numbers of $p$-adic random matrix products at both the bulk and edge. The limit object which we construct, the reflecting Poisson sea, may thus be viewed as a $p$-adic analogue of line ensembles appearing in classical random matrix theory. However, in contrast to those it is a discrete space Poisson-type particle system with only local reflection interactions and no obvious determinantal structure. The limits hold for any $\mathrm{GL}_n(\mathbb{Z}_p)$-invariant matrix distributions under weak universality hypotheses, with no spatial rescaling.

Figures (3)

• Figure 1: A sample path trajectory of $(\mathcal{S}^{\mu,2\infty}_i(T))_{i \in \mathbb{Z}}$, where the vertical direction represents space and the horizontal direction represents time, and there are infinitely many paths above and below those pictured. When $\mathcal{S}^{\mu,2\infty}_j(T) = \mathcal{S}^{\mu,2\infty}_{j+1}(T)$ we draw the paths slightly shifted so both are visible.
• Figure 2: Plot of a realization of the paths $\mathop{\mathrm{SN}}\nolimits(A_\tau \cdots A_1)_i, i = 1,2,3,4$, depicted as piecewise-constant functions on $\mathbb{R}_{\geq 0}$, where the matrices $A_j \in \mathop{\mathrm{Mat}}\nolimits_4(\mathbb{Z}_2)$ have iid entries distributed by the additive Haar measure on $\mathbb{Z}_2$ (simulated on SAGE, data as in van2020limits). As in \ref{['fig:sea_cartoon']} we show equal singular numbers by paths slightly below one another.
• Figure 3: A sample trajectory of $\mathcal{S}^{\nu,n}(T)$ as $T \geq 0$ varies, for $n=5$ and $\nu = (3,3,2,-1,-3)$. We indicate when a path's clock rings by a cross on the path, and draw two paths at the same position slightly below one another. The top two paths $\mathcal{S}^{\nu,n}_1$ and $\mathcal{S}^{\nu,n}_2$ both begin at position $3$ and remain there until $\mathcal{S}^{\nu,n}_1$'s clock rings at time $T_1 > 0$; later, $\mathcal{S}^{\nu,n}_3$ jumps to position $3$ and then has its clock ring again at time $T_2 > T_1$, but because it is blocked by $\mathcal{S}^{\nu,n}_2$, the latter path jumps instead by condition \ref{['item:interact']}.

Theorems & Definitions (116)

• Theorem 1.1
• Remark 1
• Remark 2
• Theorem 1.2
• Example 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 3
• Remark 4
• Proposition 2.1