Truncation of Haar random matrices in $\mathrm{GL}_N(\mathbb{Z}_m)$

The asymptotic law of the truncated $S\times S$ random submatrix of a Haar random matrix in $\mathrm{GL}_N(\mathbb{Z}_m)$ as $N$ goes to infinity is obtained. The same result is also obtained when $\mathbb{Z}_m$ is replaced by any commutative compact local ring whose maximal ideal is topologically closed.


Introduction
In the theory of random matrices, some particular attentions are payed recently to the asymptotic distributions of those truncated S ×S upper-left corner of a large N × N random matrices from different matrix ensembles (CUE, COE, Haar Unitary Ensembles, Haar Orthogonal Ensembles), see [6,4,2,1].
In the present paper, we consider the truncation of a Haar random matrix in GL N (Z m ) with Z m = Z/mZ. This research is motivated by its application in a forthcoming paper on the classification of ergodic measures on the space of infinite p-adic matrices, where the asymptotic law of a fixed size truncation of the Haar random matrix from the group of N ×N invertible matrices over the ring of p-adic integers is essentially used and is derived from a particular case of our main result, Theorem 3.1. Remark that the ring of p-adic integers is isomorphic to the inverse limit of the rings Z p n .

Notation
Fix a positive integer m ∈ N. Consider the ring Z m := Z/mZ. Let Z × m be the multiplicative group of invertible elements of the ring Z m . For any N ∈ N, denote by M N (Z m ) the matrix ring over Z m and denote by GL N (Z m ) the finite group of N × N invertible matrices over Z m .
Note that we have Let U N (m) denote the uniform distribution on M N (Z m ) and let µ N (m) denote the uniform distribution on GL N (Z m ). Note that µ N (m) is the normalized Haar measure of the group GL N (Z m ).
The cardinality of any finite set E is denoted by |E|.

Main result
Fix a positive integer S ∈ N. If X is a N × N matrix (in what follows, the range of the coefficients of X can vary), then we denote by X[S] the truncated upper-left S × S corner of X, i.e., Let X (N ) (m) be a random matrix sampled with respect to the normalized Haar measure of GL N (Z m ), that is, the probability distribution L(X (N ) (m)) of the random matrix X (N ) (m) satisfies L(X (N ) (m)) = µ N (m).  For any positive integer u ∈ N, we write Q u : Z → Z u = Z/uZ the quotient map. If v is another positive integer such that u divides v, then since vZ ⊂ uZ = ker(Q u ), the map Q u induces in a unique way a map Q v u : that is, for each element x ∈ Z u , the cardinality of the pre-image of x is v/u. By slightly abusing the notation, for any matrix By the prime factorization theorem, we may write in a unique way where p 1 , · · · , p s are distinct prime numbers and r 1 , · · · , r s are positive integers. By the Chinese remainder theorem, we have an isomorphism of the following two rings: Simple case: Let us first assume that in the factorization (3.3), we have s = 1. For simplifying the notation, let us write m = p r .
Writing F p for the finite field Z/pZ. We have the following characterization of GL N (Z p r ).
, then a moment of thinking allows us to write where p S(N −S) the number of choices of (X ij ) 1≤i≤S, S+1≤j≤N with coefficients in F p and N −S−1 It follows that, for any matrix W ∈ M S (Z p r ), we have We also have for any matrix W ∈ M S (Z p r ), Recall that X (N ) (m) is a random matrix sampled with respect to the Haar measure of GL N (Z m ) = GL N (Z p r ). By combining (3.2), (3 .7) and (3.8), we see that the cardinality n N (W ) := X ∈ GL N (Z p r ) : X(S) = W satisfies the relation As a consequence, for any W 1 , W 2 ∈ M S (Z p r ), the following relation holds: Since the set M S (Z m ) is finite, the above equality (3.10) implies that L(X (N ) (m)[S]) converges weakly, as N tends to infinity, to the uniform distribution U S (m) on M S (Z m ). General case: It is clear that, for any N ∈ N, the isomorphism φ defined in (3.5) induces in a natural way a ring isomorphism: The restriction of φ N on GL N (Z m ) induces a group isomorphism: In particular, if X (N ) (p r 1 1 ), · · · , X (N ) (p rs s ) are independent Haar random matrices in GL N (Z p r 1 1 ), · · · , GL N (Z p rs s ) respectively, then the random matrix φ −1 N (X (N ) (p r 1 1 ) ⊕ · · · ⊕ X (N ) (p rs s )) is a Haar random matrix in GL N (Z m ). Moreover, we have . We thus complete the proof of Theorem 3.1.

A generalization
Let F q denote the finite field with cardinality q = p n . Consider the Haar random matrix Z (N ) in GL N (F q ). Then we have Proof. By combinatoric arguments, we have a similar estimate as (3.9) and the proof of Theorem 4.1 then follows immediately. Here we omit the details.
Let (A , +, ·) be a topological commutative ring with identity which is compact, thus by assumption, the two operations +, · : A × A → A are both continuous. Assume also that A is a local ring. Recall that by local ring, we mean that A admits a unique maximal ideal. Let us denote the maximal ideal of A by m. If we denote by A × the multiplicative group of the A , then we have m = A \ A × . Moreover, let us assume that m is closed.  As a consequence, there exists a positive integer q = p n with p a prime number and n a positive integer, such that |A /m| = q and A /m ≃ F q . Let {a i : i = 0, · · · , q − 1} be a subset of A which forms a complete set of representatives of A /m, assume moreover that From now on, as a set, we will identify {a i : i = 0, · · · , q − 1} with F q . For instance, under this identification, we may write we also identify the following subset of M N (A ): Since A × is closed, indeed, the group of invertible matrices over A : as a closed subset of M N (A ), is compact. As a consequence, we may speak of Haar random matrix in GL N (A ), let Y (N ) be such a random matrix. We would like to study the asymptotic law of the truncated random matrix Y (N ) [S] as N goes to infinity. where we identify GL N (F q ) with the set given by (4.15).
Proof. It is easy to see that for any X ∈ GL N (F q ) and any X ′ ∈ M N (m), we have det(X + X ′ ) ≡ det X( mod m), hence det(X + X ′ ) ∈ A × . This implies that the set on the right hand side of (4.16) is contained in GL N (A ). Conversely, an element A ∈ GL N (A ) ⊂ M N (A ) corresponds naturally to a matrix X A ∈ M N (A ) all of whose coefficients are in F q (identified with {a i : 0 ≤ i ≤ q − 1}) such that A ≡ X A ( mod m) and det A ≡ det X A ( mod m).
As a consequence, det X A ∈ A × and hence X A ∈ GL N (F q ). This shows that GL N (A ) is contained in the set on the right hand side of (4.16).
Finally, by the definition of the set GL N (F q ) in (4.16), it is clear that all the subsets X + M N (m), X ∈ GL N (F q ) are disjoint.
As an immediate consequence of Lemma 4.3, we have the following corollary. First recall that we have identified GL N (F q ) with the set (4.15), hence the random matrix Z (N ) may be considered as a random matrix sampled uniformly from the set (4.15). Note that M N (m) ≃ m N ×N is equipped with the uniform probability Corollary 4.4. Assume that we are given a random matrix U (N ) sampled uniformly from M N (m), which is independent from the random matrix Z (N ) . The the random matrix is a Haar random matrix in GL N (A ).
Note that the distributions of the two random matrices U (S) and U (N ) [S] coincide.