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Exact cospectrality probabilities for uniform random matrices

Alexander Van Werde

TL;DR

The paper derives an exact, non-asymptotic formula for the probability that conjugating a random symmetric matrix by a fixed orthogonal matrix preserves integrality, expressed via Smith ideals of the scaled orthogonal matrix. It then applies the formula to rational cospectrality, obtaining precise low-dimensional results: for $n=2$, $\mathbb{E}[N_2(\ell)]=2^{r+3}/\ell^2$ under a specific prime-factor condition, and for $n=3$, $\mathbb{E}[N_3(\ell)]=48/\ell^2 \prod_{p\mid\ell}(1+1/p)$, with non-monotone dependence on the denominator. The method hinges on Smith normal form invariants over Dedekind domains and a translation-invariant averaging argument modulo ideals, offering a unified approach that extends beyond the specific ensemble. The results yield nontrivial bounds on the probability of rational cospectrality at fixed denominators and illuminate how arithmetic fluctuations influence spectral-conjugacy phenomena in low dimensions, with potential extensions to Hermitian and non-symmetric matrices.

Abstract

We study the conjugation action of orthogonal matrices on symmetric random matrices. Given a fixed orthogonal matrix over an algebraic number field and a random matrix with entries sufficiently uniform in the ring of integers, we wonder what the probability is that the conjugate is again integral. Our main result establishes an exact formula for this probability in terms of the Smith ideals associated to the orthogonal matrix. As an illustrative application, we establish exact formulas for the expected number of rational orthogonal matrices that preserve the integrality of a random matrix for every fixed denominator in dimensions two and three. Notably, the dependence on the denominator turns out to be non-monotone due to number-theoretic fluctuations. We also prove bounds on the probability of rational cospectrality with bounded but arbitrarily large denominator.

Exact cospectrality probabilities for uniform random matrices

TL;DR

The paper derives an exact, non-asymptotic formula for the probability that conjugating a random symmetric matrix by a fixed orthogonal matrix preserves integrality, expressed via Smith ideals of the scaled orthogonal matrix. It then applies the formula to rational cospectrality, obtaining precise low-dimensional results: for , under a specific prime-factor condition, and for , , with non-monotone dependence on the denominator. The method hinges on Smith normal form invariants over Dedekind domains and a translation-invariant averaging argument modulo ideals, offering a unified approach that extends beyond the specific ensemble. The results yield nontrivial bounds on the probability of rational cospectrality at fixed denominators and illuminate how arithmetic fluctuations influence spectral-conjugacy phenomena in low dimensions, with potential extensions to Hermitian and non-symmetric matrices.

Abstract

We study the conjugation action of orthogonal matrices on symmetric random matrices. Given a fixed orthogonal matrix over an algebraic number field and a random matrix with entries sufficiently uniform in the ring of integers, we wonder what the probability is that the conjugate is again integral. Our main result establishes an exact formula for this probability in terms of the Smith ideals associated to the orthogonal matrix. As an illustrative application, we establish exact formulas for the expected number of rational orthogonal matrices that preserve the integrality of a random matrix for every fixed denominator in dimensions two and three. Notably, the dependence on the denominator turns out to be non-monotone due to number-theoretic fluctuations. We also prove bounds on the probability of rational cospectrality with bounded but arbitrarily large denominator.
Paper Structure (10 sections, 21 theorems, 48 equations, 1 figure)

This paper contains 10 sections, 21 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Exact formula
  3. Application to rational cospectrality
  4. Related work
  5. Proof of Theorem
  6. Proofs for the applications to rational cospectrality
  7. Proof of Theorem
  8. Proof of Theorem
  9. Proof of Corollary
  10. Details for Eq. in the proof of Lemma

Key Result

Theorem 1.3

Let ${Q}\in K^{n\times n}$ be a matrix with level $\ell$ that is orthogonal. Consider a random symmetric $n\times n$ matrix ${X}\in \mathscr{O}_K^{n\times n}$ whose upper-triangular entries $\{{X}_{i,j}:i\leq j \}$ have independent and uniformly distributed reductions in $\mathscr{O}_K/\ell^2 \maths

Figures (1)

  • Figure 1: The sequence $\ell \mapsto \mathbb{E}[N_n(\ell)]$ is not monotone for $n=2,3$.

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Example 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Conjecture 1.9
  • Lemma 2.1
  • ...and 37 more