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Roger Van Peski

Publications and source records attributed to Roger Van Peski.

At least 19 recordsLinked to original sources
arXiv2026

Rank fluctuations of matrix products and a moment method for growing groups

We consider the cokernel $G_n = \mathbf{Cok}(A_{k} \cdots A_2 A_1)$ of a product of independent $n \times n$ random integer matrices with iid entries from generic nondegenerate distributions, in the regime where both $n$ and $k$ are sent to $\infty$ simultaneously. In this regime we show that the cokernel statistics converge universally to the reflecting Poisson sea, an interacting particle system constructed in arXiv:2312.11702, at the level of $1$-point marginals. In particular, $\operatorname{corank}(A_{k} \cdots A_2 A_1 \pmod{p}) \sim \log_p k$, and its fluctuations are $O(1)$ and converge to a discrete random variable defined in arXiv:2310.12275. The main difference with previous works studying cokernels of random matrices is that $G_n$ does not converge to a random finite group; for instance, the $p$-rank of $G_n$ diverges. This means that the usual moment method for random groups does not apply. Instead, we proceed by proving a `rescaled moment method' theorem applicable to a general sequence of random groups of growing size. This result establishes that fluctuations of $p$-ranks and other statistics still converge to limit random variables, provided that certain rescaled moments $\mathbb{E}[\#\operatorname{Hom}(G_n,H)]/C(n,H)$ converge.

math.PR↗
arXiv2026

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

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.

math.PR↗
arXiv2026

The rank of a random triangular matrix over $\mathbb{F}_q$

We consider uniformly random strictly upper-triangular matrices in $\operatorname{Mat}_n(\mathbb{F}_q)$. For such a matrix $A_n$, we show that $n-\operatorname{rank}(A_n) \approx \log_q n$ as $n \to \infty$, and find that the fluctuations around this limit are finite-order and given by explicit $\mathbb{Z}$-valued random variables. More generally, we consider the random partition whose parts are the sizes of the nilpotent Jordan blocks of $A_n$: its $k$ largest parts (rows) were previously shown by Borodin to have jointly Gaussian fluctuations as $N \to \infty$, and its columns correspond to differences $\operatorname{rank}(A_n^{i-1}) - \operatorname{rank}(A_n^i)$. We show the fluctuations of the columns converge jointly to a discrete random point configuration $\mathcal{L}_{t,χ}$ introduced in arXiv:2310.12275. The proofs use an explicit integral formula for the probabilities at finite $N$, obtained by de-Poissonizing a corresponding one in arXiv:2310.12275, which is amenable to asymptotic analysis.

math.PR↗
arXiv2026

Groups with pairings, Hall modules, and Hall-Littlewood polynomials

We relate the combinatorics of Hall-Littlewood polynomials to that of abelian $p$-groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian $p$-groups (without pairings) and Hall-Littlewood polynomials. Specifically, we define a module over the classical Hall algebra with basis indexed by groups with pairings, and explicitly relate its structure constants to Hall-Littlewood polynomials at different values of the parameter $t$. We also show certain expectation formulas with respect to Cohen-Lenstra type measures on groups with pairings. In the alternating case this gives a new and simpler proof of previous results of Delaunay-Jouhet.

math.CO↗
arXiv2026

Eigenvalues of $p$-adic random matrices

We develop the basic theory of eigenvalues of $p$-adic random matrices, analogous to the classical theory for random matrices over $\mathbb{R}$ and $\mathbb{C}$. Such eigenvalue statistics were proposed as a model for the zeroes of $p$-adic $L$-functions by Ellenberg-Jain-Venkatesh, who computed the limiting distribution of the number of eigenvalues in a unit disc. We compute the full joint distribution of the $n$ eigenvalues of an $n \times n$ matrix with Haar distribution, obtaining Coulomb gas type formulas as in the archimedean case, with Vandermonde terms leading to eigenvalue repulsion. From these Coulomb gas density functions we derive asymptotics of eigenvalue statistics as $n \to \infty$. These include exact computations, such as a closed form $$ρ(x,y) = 1 - θ_3(-\sqrt{p};||x-y||^2/p)$$ for the limiting pair correlation of eigenvalues in $\mathbb{Z}_p$, and similar results in quadratic extensions. Such formulas yield concrete numerical predictions on zeroes of $p$-adic $L$-functions. For eigenvalues in arbitrary extensions of $\mathbb{Q}_p$ we also give precise estimates on their pair-repulsion and expected number of eigenvalues in each extension. Finally, we compute the asymptotic probability that all eigenvalues lie in $\mathbb{Z}_p$. Our proofs combine results from several distinct areas: $p$-adic orbital integrals, roots of random $p$-adic polynomials, the Sawin-Wood moment method for random modules, and Markov chains associated with measures on integer partitions.

math.NT↗
arXiv2025

The Gamma-disordered Aztec diamond

We introduce a multi-parameter family of random edge weights on the Aztec diamond graph, given by certain Gamma variables, and prove several results about the corresponding random dimer measures. Firstly, we show there is no phase transition at the level of the free energy. This provides rigorous backing for the physics predictions of Zeng-Leath-Hwa and later works that dimer models with random weights are in the glassy `super-rough' phase at all temperatures with no phase transition. Secondly, we show that the random dimer covers themselves enjoy exact distributional equalities of certain marginals with path locations in new `hybrid' integrable polymers. These reduce to the stationary log-Gamma, strict-weak, and Beta polymer in random environment in certain cases, allowing transfer of known results from integrable polymers to dimers with random weights. As an example application, we prove that the turning points at the boundaries of the Aztec diamond exhibit fluctuations of order $n^{2/3}$, in contrast to the $n^{1/2}$ fluctuations for deterministic weights. Underlying all these is a key integrability property of the weights: they are the unique family for which independence is preserved under the shuffling algorithm.

math.PR↗
arXiv2025

Cohen-Lenstra flag universality for random matrix products

For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this flag converges universally, for any nondegenerate entry distribution, to the Cohen-Lenstra type measure which weights each flag inversely proportional to the size of its automorphism group. As a corollary, we prove universality of certain formulas for the limiting conditional distribution of $\mathrm{cok}(M_1M_2)$ given $\mathrm{cok}(M_1),\mathrm{cok}(M_2)$ in terms of Hall-Littlewood structure constants, which were previously obtained only for Haar matrices over $\mathbb{Z}_p$. Our proofs combine the general technology of Sawin-Wood, matrix product moment computations following those of Nguyen-Van Peski, and the computation done previously for Haar $p$-adic matrices by Huang.

math.PR↗
arXiv2025

Non-Archimedean GUE corners and Hecke modules

We compute the joint distribution of singular numbers for all principal corners of a $p$-adic Hermitian (resp. alternating) matrix with additive Haar distribution, the non-archimedean analogue of the GUE (resp. aGUE) corners process. In the alternating case we find that it is a Hall-Littlewood process, explaining -- and recovering as a corollary -- results of Fulman-Kaplan. In the Hermitian case we obtain a `marginal distribution' of a formal Hall-Littlewood process with both positive and negative transition `probabilities'. The proofs relate natural random matrix operations to structural results of Hironaka and Hironaka-Sato on modules over the spherical Hecke algebra, yielding other probabilistic statements of independent interest along the way.

math.PR↗
arXiv2025

On the moments of one-level densities in families of holomorphic cusp forms in the level aspect

We study the $n^{\rm th}$ centered moments of the $1$-level density for the low-lying zeros of $L$-functions attached to holomorphic cuspidal newforms of large prime level and fixed weight. Assuming the Generalized Riemann Hypotheses, we compute this statistic for any $n\ge 1$ and for all test functions whose Fourier transforms are supported in $\left(-2/n, \, 2/n\right)$. This is believed to be the natural limit of the current technology. Our work significantly extends beyond the trivial range $(-1/n, \, 1/n)$ and surpasses the previous record of $(-1/(n-1),\, 1/(n-1))$ whenever $n>2$. The Katz-Sarnak philosophy predicts that the aforementioned statistic can be modeled by the corresponding statistic for the eigenvalues of random orthogonal matrices. We prove that this is the case for test functions with Fourier support contained in $(-2/n,\, 2/n)$. The main technical innovation is a tractable vantage to evaluate the combinatorial zoo of terms, similar to the work of Conrey-Snaith and Mason-Snaith. As an application, our work provides better bounds on the order of vanishing at the central point for the $L$-functions in our family.

math.NT↗
arXiv2024

Local limits in $p$-adic random matrix theory

We study the distribution of singular numbers of products of certain classes of $p$-adic random matrices, as both the matrix size and number of products go to $\infty$ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on $\mathbb{Z}$, defined explicitly in terms of certain intricate mixed $q$-series/exponential sums. This object may be viewed as a nontrivial $p$-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg arXiv:1809.05905, which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from $\mathrm{GL}_N(\mathbb{Z}_p)$, and the $p$-adic analogue of Dyson Brownian motion studied in arXiv:2112.03725.

math.PR↗
arXiv2024

What is a $p$-adic Dyson Brownian motion?

We consider the singular numbers of a certain explicit continuous-time Markov jump process on $\mathrm{GL}_N(\mathbb{Q}_p)$, which we argue gives the closest $p$-adic analogue of multiplicative Dyson Brownian motion. We do so by explicitly classifying the possible dynamics of singular numbers of processes on $\mathrm{GL}_N(\mathbb{Q}_p)$ satisfying natural properties possessed by Brownian motion on $\mathrm{GL}_N(\mathbb{C})$. Computing the evolution of singular numbers explicitly, we find that the $N$-tuple of singular numbers in decreasing order evolves as a Poisson jump process on $\mathbb{Z}^N$, with ordering enforced by reflection off the walls of the positive type $A$ Weyl chamber. This contrasts with -- and provides a $p$-adic analogue to -- the behavior of classical Dyson Brownian motion, where ordering is enforced by conditioning to avoid the Weyl chamber walls.

math.PR↗
arXiv2023

Universality for cokernels of random matrix products

For random integer matrices $M_1,\ldots,M_k \in \operatorname{Mat}_n(\mathbb{Z})$ with independent entries, we study the distribution of the cokernel $\operatorname{cok}(M_1 \cdots M_k)$ of their product. We show that this distribution converges to a universal one as $n \to \infty$ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of $\operatorname{cok}(M_1),\operatorname{cok}(M_1M_2),\ldots,\operatorname{cok}(M_1 \cdots M_k)$. Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over $\mathbb{F}_p$ as the matrix size tends to infinity.

math.PR↗
arXiv2022

$q$-TASEP with position-dependent slowing

We introduce a new interacting particle system on $\mathbb{Z}$, \emph{slowed $t$-TASEP}. It may be viewed as a $q$-TASEP with additional position-dependent slowing of jump rates depending on a parameter $t$, which leads to discrete and nonuniversal asymptotics at large time. If on the other hand $t \to 1$ as $\text{time} \to \infty$, we prove (1) a law of large numbers for particle positions, (2) a central limit theorem, with convergence to the fixed-time Gaussian marginal of a stationary solution to SDEs derived from the particle jump rates, and (3) a bulk limit to a certain explicit stationary Gaussian process on $\mathbb{R}$, with scaling exponents characteristic of the Edwards-Wilkinson universality class in $(1+1)$ dimensions. The proofs relate slowed $t$-TASEP to a certain Hall-Littlewood process, and use contour integral formulas for observables of this process. Unlike most previously studied Macdonald processes, this one involves only local interactions, resulting in asymptotics characteristic of $(1+1)$-dimensional rather than $(2+1)$-dimensional systems.

math.PR↗
arXiv2022

Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

We prove that the boundary of the Hall-Littlewood $t$-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when $1/t$ is a prime $p$ we use this to recover results of Bufetov-Qiu and Assiotis on infinite $p$-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions. Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain a simple expression for the joint distribution of the cokernels of products $A_1, A_2A_1, A_3A_2A_1,\ldots$ of independent Haar-distributed matrices $A_i$ over the $p$-adic integers $\mathbb{Z}_p$. This expression generalizes the explicit formula for the classical Cohen-Lenstra measure on abelian $p$-groups.

math.CO↗
arXiv2022

Lozenge tilings and the Gaussian free field on a cylinder

We use the periodic Schur process, introduced in arXiv:math/0601019v1, to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the $q^{\operatorname{vol}}$ measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes.

math.PR↗
arXiv2022

Lyapunov exponents for truncated unitary and Ginibre matrices

In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced `picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on $\operatorname{GL}_n(\mathbb{C})$, analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.

math.PR↗
arXiv2021

Limits and fluctuations of $p$-adic random matrix products

We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over $\mathbb{Q}_p$ are governed by the Hall-Littlewood polynomials, in a structurally identical manner to the classical relations between singular values of complex random matrices and Heckman-Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $\text{GL}_N(\mathbb{Z}_p)$ form a discrete-time Markov chain distributed as a Hall-Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall-Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of $\text{GL}_N(\mathbb{Z}_p)$ corners.

math.PR↗