Persistence diagrams of random triangular matrices over finite fields

Abstract: Let us consider a random infinite lower triangular matrix, where the entries on and below the diagonal are i.i.d. uniform random elements of a fixed finite field. We investigate the evolution of the span of the first $n$ rows of this matrix as $n$ grows. Many properties of this evolving subspace can be captured with the help of the verbose persistence diagram, which is a standard tool in stochastic topology and topological data analysis. We give an explicit formula for the distribution of the persistence diagram. We prove a law of large numbers for the distribution of lifetimes. We also describe the fluctuations of the persistent Betti numbers.

• The paper introduces explicit probability formulas for persistence diagram configurations using combinatorial invariants and inversion numbers.
• It establishes a law of large numbers for lifetime distributions via sums of shifted geometric variables, linking persistent homology to random matrix theory.
• The work connects stochastic topology to universality phenomena, offering potential applications in topological data analysis and statistical null-models.

Persistence Diagrams of Random Triangular Matrices Over Finite Fields

Overview

The paper "Persistence diagrams of random triangular matrices over finite fields" (2606.17895) presents a rigorous probabilistic analysis of the evolution of rowspace coranks in infinite lower triangular matrices with i.i.d. uniform entries over a finite field $\mathbb{F}_q$. The work leverages tools from stochastic topology, notably verbose persistence diagrams, to systematically encode the birth and death of vectors in subspace filtrations. The author provides explicit formulas for the distributions of persistence diagrams and describes limiting behaviors of lifetimes and persistent Betti numbers, connecting the study to universality phenomena in random matrix theory.

Mathematical Framework and Main Results

The central object is an infinite random lower triangular matrix $L$ with entries from $\mathbb{F}_q$, focusing on the sequence of subspaces $\mathrm{RowSpace}(L_n)$ as $n$ grows. The persistence diagram $P$ records for each vector $v\in \mathbb{F}_q^{\mathbb{Z}_+}$ a birth time $b(v)$ (minimal $n$ so $v\in Z_n$) and a death time $L$0 (minimal $L$1 so $L$2), classifying the vector evolution through the filtration.

Explicit Distribution Formulas

An explicit probability formula (Theorem 2) is derived for the occurrence of admissible subsets $L$3 in the persistence diagram $L$4, in terms of combinatorial invariants—most notably the inversion number $L$5 and an energy-like quantity $L$6—with weights governed by powers of $L$7. This formula quantifies the structure and randomness of births and deaths in the matrix process.

Lifetimes and Law of Large Numbers

Theorem 3 establishes a law of large numbers for the lifetime distribution (death-birth) of points in the persistence diagram. The limiting lifetime distribution is expressed via sums of independent shifted geometric random variables, parameterized by $L$8. As $L$9, the empirical frequency of vectors with fixed lifetime converges to a theoretically predicted probability, revealing precise temporal homological evolution in random filtrations.

Fluctuations of Persistent Betti Numbers

The persistent Betti numbers $\mathbb{F}_q$0 are analyzed in various scaling regimes (Theorem 4). Four distinct phases are isolated, based on growth rates of $\mathbb{F}_q$1 relative to $\mathbb{F}_q$2, with limiting distributions involving universal random variables such as $\mathbb{F}_q$3 and distributions studied in p-adic random matrix theory. Notably, connections are drawn to the Cohen-Lenstra distribution and its universality class.

Algorithmic Construction and Sampling

A Monte-Carlo-like procedure, NextLine and its generalizations, is proposed for sampling persistence diagrams, leveraging coin tosses with success probability $\mathbb{F}_q$4 and Markov chain transitions. The theoretical analysis demonstrates how the algorithm mirrors the evolution of the matrix's kernel and establishes the independence and distributional properties required for finite and infinite matrix sampling.

Connections to Homology and Universality

The author situates the work within broader contexts:

• Persistent homology of growing simplicial complexes (e.g., Čech complexes) is shown to be analogous to the persistence module structure of random lower triangular matrices.
• Universality results from the random matrix cokernel literature, especially those tied to Cohen-Lenstra heuristics, are linked to the persistence diagram's limiting distributions.
• The approach bridges gaps between stochastic topology (persistent homology, Betti numbers) and dynamical random matrix theory (time evolution of cokernels, limiting distributions).

Numerical and Structural Highlights

• The explicit probability formula for persistence diagram configurations is derived, involving nontrivial combinatorial and algebraic quantities.
• A law of large numbers for the distribution of lifetimes is established, with limiting probabilities determined precisely by sums of geometric variables.
• Persistent Betti numbers exhibit phase transitions based on growth rates, with sharp limits described in terms of universal random variables and exponential type distributions closely related to the Cohen-Lenstra heuristics.
• The evolution of kernel dimension is connected to deep universality phenomena, extending prior results on random matrix cokernels and group structures.

Implications and Future Directions

From a theoretical standpoint, the paper advances the algebraic and probabilistic understanding of persistence modules arising from random matrix filtrations. It provides rigorous probabilistic tools to quantify not just instantaneous spectral statistics but their full dynamical evolution, encoded in persistence diagrams.

Practically, the explicit algorithms and formulas can potentially operate as statistical baselines or null-models for assessing persistent homology in high-throughput topological data analysis settings, especially as these often utilize filtrations over finite fields. The universality class elucidated in the random matrix context suggests robust behaviors valid across broad model classes, including block matrix and matrix product scenarios.

Future work may include:

• Extension to other structured random matrix models, e.g., block-triangular, Toeplitz, or sparse matrices, leveraging the persistent module approach.
• Analysis of joint distributional fluctuations for finite collections of persistent Betti numbers, as posed in the paper's open problems.
• Direct application in statistical topology, e.g., evaluating topological features from random data clouds or network models.
• Investigation of the connections to algebraic number theory, such as direct analogs to class group statistics and p-adic matrix universality.

Conclusion

This work introduces a formal, combinatorial and probabilistic machinery for understanding persistence diagrams in the setting of random lower triangular matrices over finite fields. The results elucidate not just individual evolutionary statistics (ranks, cokernels) but the joint temporal structure via persistence diagrams, offering explicit formulas and asymptotic results tied to universality classes in random matrix theory. The connection to stochastic topology and the use of persistence modules mark a significant step in bridging algebraic and probabilistic perspectives on matrix filtration dynamics.