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András Mészáros

Publications and source records attributed to András Mészáros.

At least 19 recordsLinked to original sources
arXiv2026

The rank evolution of block bidiagonal matrices over finite fields

We investigate uniform random block lower bidiagonal matrices over the finite field~$\mathbb{F}_q$, and prove that their corank undergoes a phase transition. First, we consider block lower bidiagonal matrices with $(k_n+1)\times k_n$ blocks where each block is of size $n\times n$. We prove that if $k_n\ll q^{n/2}$, then the corank of these matrices is $0$ with high probability, and if $k_n\gg q^{n/2}$, then the corank has Gaussian fluctuations. Second, we consider block lower bidiagonal matrices with $k_n\times k_n$ blocks where each block is of size $n\times n$. We prove that if $k_n\ll q^{n/2}$, then the corank exhibits the same constant order fluctuations as the corank of the matrix products considered by Nguyen and Van Peski, and if $k_n\gg q^{n/2}$, then the corank has Gaussian fluctuations. Finally, we also consider a truncated version of the first model, where we prove that at $k_n\approx q^{n/2}$, we have a phase transition between a Cohen-Lenstra and a Gaussian limiting behavior of the corank. We also show that there is a localization/delocalization phase transition for the vectors in the kernels of these matrices at the same critical point. In all three cases, we also provide a precise description of the behavior of the corank at criticality.

math.PR↗
arXiv2026

Product approximations for uniform spanning trees and determinantal processes

We prove that for every finite connected simple graph $G$ on $n$ vertices with minimum degree $k\geq 2$, there is a coupling of its uniform spanning tree and its random $1$-out subgraph under which the expected number of differing edges is $O(n\log (k)/\sqrt{k})$. More generally, we establish analogous couplings between a wide class of determinantal processes and their corresponding $1$-out processes, including Kalai's determinantal hypertrees, with expected symmetric difference negligible compared to the size of the determinantal set. While the graphical case admits a surprisingly short and elegant proof, the more general result requires establishing an asymptotic equipartition property for the determinantal measures in this class, from which we also derive exponential growth rates for the homology torsion of determinantal hyperforests in higher-dimensional regular high-degree complexes.

math.PR↗
arXiv2026

Cocycles of determinantal hypertrees with small support

Let ${T}_n$ be a random $2$-dimensional determinantal hypertree on $n$ vertices. Given any prime $p$, we answer the following question: If a cocycle in $Z^1({T}_n,\mathbb{F}_p)$ has small support, what does the support typically look like? More precisely, we characterize all the finite connected graphs $G$ for which there is a constant $c_G>0$ with the following property: For all large enough $n$, with probability at least $c_G$, we have a cocycle $f\in Z^1({T}_n,\mathbb{F}_p)$ such that after removing all the isolated vertices, the support of $f$ is isomorphic to $G$. We prove that for $p>2$, we do not have any such graph. For $p=2$, a connected graph has the property above if and only if it has a unique cycle such that this unique cycle has odd length, moreover, if the unique cycle is a triangle, then we also need to require that all the vertices of the triangle have degree at least $3$.

math.CO↗
arXiv2026

Persistence diagrams of random triangular matrices over finite fields

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.

math.CO↗
arXiv2026

The homology torsion growth of determinantal hypertrees

Fix a dimension $d\ge 2$, and let $T_n$ be a random $d$-dimensional determinantal hypertree on $n$ vertices. We prove that \[\frac{\log|H_{d-1}(T_n,\mathbb{Z})|}{n\choose {d}}\] converges in probability to a constant $c_d$, which satisfies \[\frac{1}2 \log\left(\frac{d+1}e\right)\le c_d\le \frac{1}2 \log\left(d+1\right) .\]

math.CO↗
arXiv2025

The persistent homology of the Linial-Meshulam process

For a fixed dimension $k\ge 1$, let us consider the randomly growing simplical complex on the vertex set $\{1,2,\dots,n\}$ defined as follows: We start with the empty complex, and for each $k+1$-element subset $σ$ of $\{1,2,\dots,n\}$, we add $σ$ and all of its subsets to the complex at some random time $t_σ$, where $(t_σ)$ are i.i.d. uniform random elements of $[0,n]$. As the complex evolves, new $k-1$-dimensional cycles are born and then at a later time they die, that is, they get filled in. The notion of persistence diagrams, which is a standard tool in topological data analysis, provides a way to record these birth and death times. In this paper, we understand the asymptotic behavior of the persistence diagrams of the above defined randomly evolving complexes as $n$ goes to infinity. As the single time marginals of the above process are variants of the Linial-Meshulam complex, our results can be viewed as extensions of the results of Linial and Peled on the Betti numbers of the Linial-Meshulam complex. Our proof relies on the notion of local weak convergence of graphs and a generalization of the results of Bordenave, Lelarge and Salez on the rank of sparse random matrices.

math.PR↗
arXiv2025

Universal constant order fluctuations for the cokernels of block triangular matrices

We prove that for a large class of random block lower triangular matrices, the Sylow $p$-subgroups of their cokernels have the same constant order fluctuations as that of the matrix products studied by Nguyen and Van Peski in arXiv:2409.03099. We also show that the theorem of Nguyen and Van Peski remains true under a weaker assumption on the number of factors in the matrix products.

math.PR↗
arXiv2025

Using dense graph limit theory to count cocycles of random simplicial complexes

We develop a limit theory for $1$-cochains of complete graphs with coefficients from a finite abelian group. We prove an analogue of the large deviation principle of Chatterjee and Varadhan for random cochains. We use these new tools to prove results about the homology of random $2$-dimensional simplicial complexes. More specifically, we prove that if $T_n$ is a random $2$-dimensional determinantal hypertree on $n$ vertices and $p$ is any prime, then \[\frac{\dim H_1(T_n,\mathbb{F}_p)}{n^2}\] converges to zero in probability. The same result holds for random $1$-out 2-complexes.

math.CO↗
arXiv2025

The fluctuations of the mod p rank of triangular matrices

We consider random lower triangular matrices such that the entries on and below the diagonal are i.i.d. copies of some $\mathbb{Z}$-valued random variable. We prove that the Sylow $p$-subgroups of the cokernels of these matrices have the same constant order fluctuations as that of the matrix products studied by Nguyen and Van Peski. As a special case, we can describe the limiting fluctuations of the rank of lower triangular matrices over $\mathbb{F}_p$ with i.i.d. random entries on and below the diagonal.

math.PR↗
arXiv2025

Last passage percolation in a product-type random environment

We consider a last passage percolation model in dimension $1+1$ with potential given by the product of a spatial i.i.d. potential with symmetric bounded distribution and an independent i.i.d. in time sequence of signs. We assume that the density of the spatial potential near the edge of its support behaves as a power, with exponent $κ>-1$. We investigate the linear growth rate of the actions of optimal point-to-point lazy random walk paths as a function of the path slope and describe the structure of the resulting shape function. It has a corner at $0$ and, although its restriction to positive slopes cannot be linear, we prove that it has a flat edge near $0$ if $κ>0$. For optimal point-to-line paths, we study their actions and locations of favorable edges that the paths tends to reach and stay at. Under an additional assumption on the time it takes for the optimal path to reach the favorable location, we prove that appropriately normalized actions converge to a limiting distribution that can be viewed as a counterpart of the Tracy-Widom law. Since the scaling exponent and the limiting distribution depend only on the parameter $κ$, our results provide a description of a new universality class.

math.PR↗
arXiv2025

Bounds on the mod 2 homology of random 2-dimensional determinantal hypertrees

As a first step towards a conjecture of Kahle and Newman, we prove that if $T_n$ is a random $2$-dimensional determinantal hypertree on $n$ vertices, then \[\frac{\dim H_1(T_n,\mathbb{F}_2)}{n^2}\] converges to zero in probability. Confirming a conjecture of Linial and Peled, we also prove the analogous statement for the $1$-out $2$-complex. Our proof relies on the large deviation principle for the Erdős-Rényi random graph by Chatterjee and Varadhan.

math.CO↗
arXiv2024

Cohen-Lenstra distribution for sparse matrices with determinantal biasing

Let us consider the following matrix $B_n$. The columns of $B_n$ are indexed with $[n]=\{1,2,\dots,n\}$ and the rows are indexed with $[n]^3$. The row corresponding to $(x_1,x_2,x_3)\in [n]^3$ is given by $\sum_{i=1}^3 e_{x_i}$, where $e_1,e_2,\dots,e_n$ is the standard basis of $\mathbb{R}^{[n]}$. Let $A_n$ be random $n\times n$ submatrix of $B_n$, where the probability that we choose a submatrix $C$ is proportional to $|\det(C)|^2$. Let $p\ge 5$ be a prime. We prove that the asymptotic distribution of the $p$-Sylow subgroup of the cokernel of $A_n$ is given by the Cohen-Lenstra heuristics. Our result is motivated by the conjecture that the first homology group of a random two dimensional hypertree is also Cohen-Lenstra distributed.

math.PR↗
arXiv2024

The local weak limit of $k$-dimensional hypertrees

Let $\mathcal{C}(n,k)$ be the set of $k$-dimensional simplicial complexes $C$ over a fixed set of $n$ vertices such that: (1) $C$ has a complete $k-1$-skeleton; (2) $C$ has precisely ${{n-1}\choose {k}}$ $k$-faces; (3) the homology group $H_{k-1}(C)$ is finite. Consider the probability measure on $\mathcal{C}(n,k)$ where the probability of a simplicial complex $C$ is proportional to $|H_{k-1}(C)|^2$. For any fixed $k$, we determine the local weak limit of these random simplicial complexes as $n$ tends to infinity. This local weak limit turns out to be the same as the local weak limit of the $1$-out $k$-complexes investigated by Linial and Peled.

math.CO↗
arXiv2024

The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra

Let $T_n$ be a $2$-dimensional determinantal hypertree on $n$ vertices. Kahle and Newman conjectured that the $p$-torsion of $H_1(T_n,\mathbb{Z})$ asymptotically follows the Cohen-Lenstra distribution. For $p=2$, we disprove this conjecture by showing that given a positive integer $h$, for all large enough $n$, we have \[\mathbb{P}(\dim H_1(T_n,\mathbb{F}_2)\ge h)\ge \frac{e^{-200h}}{(100h)^{5h}}.\] We also show that $T_n$ is a bad cosystolic expander with positive probability.

math.CO↗
arXiv2023

Eigenvectors of the square grid plus GUE

Eigenvectors of the GUE-perturbed discrete torus with uniform boundary conditions retain some product structure for small perturbations but converge to discrete Gaussian waves for large perturbations. We determine where this phase transition happens.

math.PR↗
arXiv2020

Matchings on trees and the adjacency matrix: A determinantal viewpoint

Let $G$ be a finite tree. For any matching $M$ of $G$, let $U(M)$ be the set of vertices uncovered by $M$. Let $\mathcal{M}_G$ be a uniform random maximum size matching of $G$. In this paper, we analyze the structure of $U(\mathcal{M}_G)$. We first show that $U(\mathcal{M}_G)$ is a determinantal process. We also show that for most vertices of $G$, the process $U(\mathcal{M}_G)$ in a small neighborhood of that vertex can be well approximated based on a somewhat larger neighborhood of the same vertex. Then we show that the normalized Shannon entropy of $U(\mathcal{M}_G)$ can be also well approximated using the local structure of $G$. In other words, in the realm of trees, the normalized Shannon entropy of $U(\mathcal{M}_G)$ -- that is, the normalized logarithm of the number of maximum size matchings of $G$ -- is a Benjamini-Schramm continuous parameter. We show that $U(\mathcal{M}_G)$ is a determinantal process through establishing a new connection between $U(\mathcal{M}_G)$ and the adjacency matrix of $G$. This result sheds a new light on the well-known fact that on a tree, the number of vertices uncovered by a maximum size matching is equal to the nullity of the adjacency matrix. Some of the proofs are based on the well established method of introducing a new perturbative parameter, which we call temperature, and then define the positive temperature analogue of $\mathcal{M}_G$, the so called monomer-dimer model, and let the temperature go to zero.

math.CO↗