Time-Inhomogeneous Random Walks on Finite Groups and Cokernels of Random Integer Block Matrices

The work in this paper is motivated by a question in the theory of integer random matrices but is of independent interest to the study of random walks on groups.

Random walks on finite groups are well-studied in the reversible, time-homogeneous, ergodic regime, where the random walk on a group $G$ consists of a product $X_{1}X_{2}\dots X_{n}$ for i.i.d. $X_{i}$ drawn from a distribution supported on a generating set of $G$. Such random walks are known to converge to the uniform distribution $\pi$ on $G$ exponentially quickly. Namely, if we denote by $\nu_{n}$ the distribution of $X_{1}X_{2}\dots X_{n}$, then

where $\sigma$ is the second-largest singular value of the Markov operator of the random walk and $||\cdot||_{L^{2}}$ denotes the $L^{2}$ norm of signed measures as functions $G\to\mathbb{R}$. See [Sal04] for an excellent review of these kinds of walks.

The above inequality comes from looking at norms of convolution operators on the space of signed measures on $G$. If $X$ and $Y$ are random elements of $G$ distributed according to $\mu$ and $\nu$ respectively, then $XY$ is distributed according to the convolution

In particular, if the $X_{i}$ are distributed according to $\mu$, then $\nu_{n}$ is the $n$-fold convolution $\mu^{*n}=\underbrace{\mu*\dots*\mu}_{n\text{ times}}$.

Some of the assumptions can be relaxed; for instance, Saloff-Coste and Zúñiga [SZ07] studied convergence of time-inhomogeneous Markov chains, including random walks on finite groups, in the case where each step of the random walk is irreducible (in particular, supported on a generating set of $G$). In that case, if we denote by $\sigma_{i}$ the second-largest singular value of the $i$th step,

The condition that each step of the random walk is supported on a generating set is crucial because if the subgroup generated by the supports of the steps is a proper subgroup of $G$, the random walk will surely stay in that subgroup. Nevertheless, one may expect that if the supports of all steps taken together generate $G$, the random walk might still equilibrate to the uniform distribution on $G$.

A consequence of our first main result is the following theorem, which relaxes this “generating” assumption by extending part of [SZ07, Theorem 3.5] to some time-inhomogeneous random walks where the probability measures driving each step need not be irreducible:

We prove a more general version of this result in Theorem 3.1.

In particular, if a time-inhomogeneous random walk on a finite group has steps supported on enough subgroups, then it converges to the uniform distribution on the group with an exponential rate controlled by subgroups that appear infrequently or mix very slowly. Adding more probability measures to the convolution $\nu_{n}$ may not improve the convergence rate, but it never makes the bound worse because convolution with a probability measure is non-expansive in the $L^{2}$ norm. A nice consequence of this is that $\mathcal{S}$ need not be an exhaustive list of every normal subgroup for which $I_{H}$ is nonempty.

The main difference between this result and [SZ07, Theorem 3.5] is that [SZ07] relaxes the time-homogeneity assumption for random walks but not the assumption that each step is supported on a generating set for the group. The new condition that the supports of the steps jointly generate $G$ is substantially weaker than the assumption that the support of each step generates $G$.

The conditions of Theorem 1.1 can be weakened so that not all the subgroups $H_{i}$ need to be normal (see Theorem 3.1), but see Example 3.3 for why some hypothesis on the subgroups is necessary.

Our main interest in developing this theorem is an application to limiting distributions of cokernels of random matrices. Wood [Woo19, Theorem 2.9] and Nguyen and Wood [NW22, Theorem 1.1] showed that cokernels of integer-valued random matrices approach a universal limiting distribution in the following sense. Let $(M_{n})_{n=1}^{\infty}$ be a sequence with each $M_{n}$ a random $n\times(n+u)$ integer matrix with independent entries ($u\geq 0$). Wood showed that, under very weak conditions on the distributions of the entries of the $M_{n}$, the distribution of the isomorphism class of the random group $\operatorname{coker}(M_{n})\coloneqq\mathbb{Z}^{n}/M_{n}(\mathbb{Z}^{n+u})$ converges weakly (i.e., at any finite collection of primes) as $n\to\infty$ to the distribution $\lambda_{u}$ on isomorphism classes of profinite abelian groups defined as follows: if $A\sim\lambda_{u}$ and $B$ is a finite abelian $p$-group, then

independently for all primes $p$, where $A_{p}$ is the $p$-part of $A$. If further $u\geq 1$, then $\lambda_{u}$ is supported on isomorphism classes of finite abelian groups, and for finite abelian $B$ we have

where $\zeta$ denotes the Riemann zeta function. Nguyen and Wood weakened the conditions on the entries and showed strong (pointwise) convergence to $\lambda_{u}$. The phenomenon that the limiting distribution of $\mathbb{Z}^{n}/M_{n}(\mathbb{Z}^{n+u})$ is rather insensitive to the distributions of the entries of $M_{n}$ is an example of universality. The distributions $\lambda_{0}$ and $\lambda_{1}$ are known as the Cohen-Lenstra distributions, and are conjectured to describe the distributions of class groups of imaginary and real quadratic number fields, respectively.

In her 2022 ICM talk, Wood [Woo23, Open Problem 3.10] asks if the universality class of $\lambda_{u}$ can be extended to cokernels of matrices with some dependent entries. There are a few specific results in this direction. Most recently, Nguyen and Wood [NW22, Theorem 1.1] show that the distribution $\lambda_{1}$ is universal for Laplacians of Erdős-Rényi random directed graphs. Mészáros [Més20] shows that $\lambda_{0}$ is universal for Laplacians of random regular directed graphs. Friedman and Washington [FW89] show that the cokernels of the random matrices $I-M$, where $M$ is drawn at random from the multiplicative Haar measure on $\operatorname{GL}_{2g}(\mathbb{Z}_{p})$, approach the $p$-part of $\lambda_{0}$ as $g\to\infty$. However, when there is too much dependence in the entries of the random matrices, one gets different (but often related) limiting distributions, for example in the case of symmetric matrices ([Woo14]), Laplacians of random regular undirected graphs ([Més20]), products of independent random integral matrices ([NV24]), and quadratic polynomials in Haar-random matrices ([CK24]). It is natural to ask just how much (and what kind of) dependence is allowed between the entries of sequences of random matrices before their cokernels leave the universality class of $\lambda_{u}$.

The main application of Theorem 3.1 in this paper is a Theorem 1.2 below, which extends the result of [Woo19] to matrices with some dependence in their rows and columns. We introduce a regularity condition on matrices, $(w,h,\varepsilon)$-balanced, in Definitions 4.2 and 4.7. Generally, it means that the matrix can be written as a block matrix where the blocks have height at most $h$, width at most $w$, are all independent, and each satisfy some regularity condition depending on $\varepsilon$. The key detail is that the blocks of the matrix may have dependent entries, as long as there is no dependence between blocks. (The $(w,h,\varepsilon)$-balanced condition is invariant under permutation of rows and columns, so one can also think of a $(w,h,\varepsilon)$-balanced matrix as a block matrix which is at most $h$ blocks tall, at most $w$ blocks wide, and such that the entries of each block are independent of each other.) With this condition, we have:

The key idea of the proof of Theorem 1.2 uses the moment method developed in [Woo14] and [Woo19]. Understanding the cokernel of a random integer matrix reduces to finding the probability that each random column maps to zero under an arbitrary surjective group homomorphism $f\colon\mathbb{Z}^{n}\to G$ for an arbitrary abelian group $G$. To handle dependent columns, we treat several columns at a time and look at the induced surjection $(\mathbb{Z}^{n})^{m}\to G^{m}$. We view the image of a random element of $(\mathbb{Z}^{n})^{m}$ as a random walk in $G^{m}$ and apply Theorem 3.1 to approximate the distribution of this image. Since the surjection $f$ is arbitrary, we have very little control over the distribution of the steps of this walk. In particular, they are almost never supported on all of $G^{m}$, which is why we need Theorem 3.1 to handle random walk steps supported on proper subgroups. The $(w,h,\varepsilon)$-balanced condition allows us to bound the singular values of the associated convolution operators and get quantitative bounds on the error in terms of $w$, $h$, and $\varepsilon$.

There is a considerable body of literature pertaining to random matrices with complex entries, with analogous universality results about distributions of eigenvalues. If $\{M_{n}\}$ is a sequence of $n\times n$ random complex matrices whose entries are independent, with appropriately normalized mean and variance, the empirical distribution of the eigenvalues of $M_{n}$ converges to the circular law, which is the uniform distribution on the unit disc in $\mathbb{C}$ [TVK10]. The universality of the circular law for spectra of a wide class of random complex block matrices was proved by Nguyen and O’Rourke in [NO15].

For a finite set $S$ and $p>0$, we use $L^{2}(S)$ to denote the space of signed measures (equivalently, real-valued functions) on $S$, equipped with the norm $||f||_{L^{2}(S)}^{2}=\sum_{s\in S}|f(s)|^{2}$. When the set $S$ is implicit, we write $||f||_{L^{2}}$ for $||f||_{L^{2}(S)}$. Any set map $f\colon S\to T$ defines a pushforward map $f_{*}\colon L^{2}(S)\to L^{2}(T)$ by $f_{*}\mu(t)=\mu(f^{-1}(t))$. We say a signed measure $\nu$ is uniform on $T\subset S$ if $\nu(t)=\nu(t^{\prime})$ for $t,t^{\prime}\in T$. For a point $f\in\mathbb{R}^{n}$ with the Euclidean metric and a linear subspace $W\subset\mathbb{R}^{n}$, we write $d_{L^{2}}(f,W)$ for the distance between $f$ and its orthogonal projection onto $W$. Note that this is equal to $\inf_{g\in W}|f-g|$. If $G$ is a finite group, any signed measure $\mu$ defines a linear convolution operator (or, if $\mu$ is a probability measure, Markov operator) $*\mu$ on $L^{2}(G)$ given by $\nu\mapsto\nu*\mu$. When we discuss the second-largest singular value of an operator, we are counting with multiplicity; for example, if the singular values of $M$ are $1,1,0$, then its second-largest singular value is $1$.

For two finite or profinite groups $G,G^{\prime}$, we write $\operatorname{Hom}(G,G^{\prime})$ for the set of (continuous) group homomorphisms from $G$ to $G^{\prime}$ and $\operatorname{Sur}(G,G^{\prime})$ for the set of (continuous) surjective group homomorphisms from $G$ to $G^{\prime}$. For a subset $S\subseteq G$, we denote by $\langle S\rangle$ the (closed) subgroup of $G$ generated by $S$. We refer to the identity element of a group as $e$.

A probability measure or distribution is a measure with total mass 1 (not signed). The uniform distribution on $G$ is usually denoted $\pi$ and is the measure on $G$ with $\pi(g)=1/|G|$ for $g\in G$. We use $\mathbb{P}[\cdot]$ for probability and $\mathbb{E}[\cdot]$ for expectation. We denote by $\operatorname{supp}\mu$ the support of a measure $\mu$. If a random variable $X$ has law $\mu$, we write $X\sim\mu$.

For a positive integer $n$, we write $[n]$ for the set $\{1,\dots,n\}$.

This section is devoted to proving the following stronger version of Theorem 1.1 from the introduction:

In the case where $k=1$ and $H_{1}=G$, we recover the first part of [SZ07, Theorem 3.5]. We postpone the proof that Theorem 3.1 implies Theorem 1.1 until the end of this section.

The following example gives a case that is covered by Theorem 3.1 but not by Theorem 1.1:

Although Theorem 3.1 makes weaker assumptions on the subgroups than Theorem 1.1, it is not possible to fully remove the normality assumption, as the following example shows:

Consider the space $\mathcal{M}=L^{2}(G)$ of $\mathbb{R}$-valued functions on a finite group $G$. Since $G$ is finite, $\mathcal{M}\cong\mathbb{R}^{G}$ (with the Euclidean norm). Let $\mathcal{M}_{0}=\{\nu\in\mathcal{M}\mid\nu(G)=0\}$. Let $\mathcal{P}\subseteq\mathcal{M}$ be the set of signed measures $\nu$ on $G$ with $\nu(G)=1$. Note that $\mathcal{P}$ and $\mathcal{M}$ are parallel affine hyperplanes in $\mathbb{R}^{G}$. Probability measures on $G$ are points in the simplex formed by the part of $\mathcal{P}$ in the positive orthant. The orthogonal complement to $\mathcal{M}_{0}$ is the line spanned by the uniform probability measure $\pi$ on $G$, and $\operatorname{span}\{\pi\}$ intersects $\mathcal{P}$ at $\pi$ and nowhere else.

Any measure $\mu_{i}$ on $G$ acts by on $\mathcal{M}$ by convolution on the right. If $\mu_{i}$ is a probability measure, the convolution operator $M_{i}\colon\nu\mapsto\nu*\mu_{i}$ also sends $\mathcal{P}$ into itself. The following lemma tells us that, in this case, $M_{i}$ contracts the distance between points of $\mathcal{P}$ and $\pi$.

The second part of Lemma 3.4 implies that if the support of $\mu_{i}$ generates $G$ and $\nu$ is a probability measure, then $||M_{i}\nu-\pi||_{L^{2}}\leq\sigma_{i}||\nu-\pi||_{L^{2}}$, so applying $M_{i}$ to a probability measure $n$ times contracts the distance to $\pi$ by a factor of $\sigma_{i}^{n}$. More generally, if the support of each $\mu$ generates $G$, then applying any combination of $M_{i}$ in any order contracts the distance to $\pi$ by the appropriate product of $\sigma_{i}$ factors. However, when the support of $\mu_{i}$ is contained in a proper subgroup of $G$, the second-largest singular value of $M_{i}$ as an operator on $\mathcal{M}$ is always 1. In this case, Lemma 3.4(2) applied to $G$ gives no useful information.

The key idea in the proof of Theorem 3.1 is that even though we cannot say outright that applying the operator $M_{i}$ moves a probability measure closer to uniform, we will show in Lemma 3.8 that $M_{i}$ moves a probability measure closer to some (explicit) subspace of $\mathcal{M}$ containing $\pi$. This subspace depends on the subgroup generated by the support of $\mu_{i}$. If we choose enough subspaces that their intersection is just $\{\pi\}$, then we will be able to show that successive application of different $M_{i}$’s moves a probability measure closer to that intersection, that is, to the uniform probability measure. The condition that our chosen subspaces intersect in $\{\pi\}$ is exactly the condition $G=\langle\bigcup_{H\in S}H\rangle$ in Theorem 1.1, or the condition that $G_{k}=\{e\}$ in Theorem 3.1.

For each subgroup $H\leq G$, let $\mathcal{M}_{H}\subseteq\mathcal{M}$ be the space of functions on $\mathcal{M}$ which are uniform on each left coset of $H$ (i.e., for $\nu\in\mathcal{M}_{H}$ and $g_{1},g_{2}\in G$ with $g_{1}^{-1}g_{2}\in H$, $\nu(g_{1})=\nu(g_{2})$).

Note that if $H$ is a subgroup of $G$ and $P\colon G\twoheadrightarrow G/H$ is the projection onto the left coset space, then one can identify $L^{2}(G/H)$ with $\mathcal{M}_{H}$ as follows:

Identifying $L^{2}(G/H)$ with $\mathcal{M}_{H}$ will allow us to prove Theorem 3.1 by induction. Using Lemma 3.8, we can say that a random walk approaches the subspace $\mathcal{M}_{H}$. Then, we can consider its projection onto $\mathcal{M}_{H}$ as a random walk on $G/H$. This allows us to ignore all random walk steps supported in $H$. The key ingredient that allows us to combine Lemma 3.8 with the inductive hypothesis is the following lemma: