Narrow Results

We prove that every profinite $n$-ary group $(G,f)={\mathrm{\text{der}}}_{𝜃,b}(G,•)$ has a unique Haar measure $m_p$ and further for every measurable subset $A\subseteq G$, we have

We study the integrals of type , depending on a matrix a ∈ M_{p × q}(ℕ), whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case a ∈ M_{2 × q}(2ℕ) to the general case a ∈ M_{p × q}(ℕ), (2) the construction of the "best algebraic normalization" of I(a), in the case a ∈ M_{2 × q}(ℕ), (3) an explicit formula for I(a), for diagonal matrices a ∈ M_{3 × 3}(ℕ), (4) a modeling result in the case a ∈ M_{1 × 2}(ℕ), in relation with the Euler–Rodrigues formula. Most proofs use various combinatorial techniques.

In this paper, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before — faithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism — for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this paper forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum groupoids to operator-algebraic completions, which both will be studied in separate papers.

Rosłanowski and Shelah [Small-large subgroups of the reals, Math. Slov. 68(3) (2018) 473–484] asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with $ZFC$ that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper, we give affirmative answers for the general case.

A short proof utilizing dynamical systems techniques is given of a necessary and sufficient condition for the normalized occupation measure of a Lévy process in a metrizable compact group to be asymptotically uniform with probability one.

Let ℤ_tr be the ring of totally real integers, Gal(ℚ) the absolute Galois group of ℚ, and e a positive integer. For each σ = (σ₁,…,σ_e) ∈ Gal(ℚ)^e let ℤ_tr(σ) be the fixed ring in ℤ_tr of σ₁,…,σ_e. Then, the theory of all first order sentences θ that are true in ℤ_tr(σ) for almost all σ ∈ Gal(ℚ)^e (in the sense of the Haar measure) is undecidable.

The line of investigation of the present paper goes back to a classical work of W. H. Gustafson of the 1973, in which it is described the probability that two randomly chosen group elements commute. In the same work, he gave some bounds for this kind of probability, providing information on the group structure. We have recently obtained some generalizations of his results for finite groups. Here we improve them in the context of the compact groups.

We prove that for the eigenvectors matrix of a Wigner matrix, under some moments conditions, the bivariate random process

We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=\alpha$, the empirical spectral measure is well approximated by a deterministic measure ${\mu }_{\alpha }$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance $d_{\mathrm{\text{BL}}}$ between the empirical spectral measure and the corresponding ${\mu }_{\alpha }$ is typically of order $\sqrt{\frac{log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.

For each n, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $x_{n,1},\dots ,x_{n,m}$ denote orthogonal nonrandom unit vectors in ${ℂ}^n$ and let $u_{n,k}={(u_k^1,\dots ,u_k^n)}^{\ast }=U_n^{\ast }{x}_{n,k}$, $k=1,\dots ,m$. Define the following functions on $[0,1]$: $X_n^{k,k}(t)=\sqrt{n}{\sum }_{i=1}^{[nt]}(|u_k^i{|}^2-\frac{1}{n})$, $X_n^{k,k^{\prime }}(t)=\sqrt{2n}{\sum }_{i=1}^{[nt]}{ū}_k^i{u}_{k^{\prime }}^i$, $k<k^{\prime }$. Then it is proven that $X_n^{k,k},\Re X_n^{k,k^{\prime }}$, $\Im X_n^{k,k^{\prime }}$, considered as random processes in $D[0,1]$, converge weakly, as $n\to \infty$, to $m^2$ independent copies of Brownian bridge. The same result holds for the $m(m+1)/2$ processes in the real case, where $O_n$ is real orthogonal Haar distributed and $x_{n,i}\in {ℝ}^n$, with $\sqrt{n}$ in $X_n^{k,k}$ and $\sqrt{2n}$ in $X_n^{k,k^{\prime }}$ replaced with $\sqrt{\frac{n}{2}}$ and $\sqrt{n}$, respectively. This latter result will be shown to hold for the matrix of eigenvectors of $M_n=(1/s)V_n{V}_n^T$ where $V_n$ is $n\times s$ consisting of the entries of $\{v_{ij}\},i,j=1,2,\dots$, i.i.d. standardized and symmetrically distributed, with each $x_{n,i}=\{\pm 1/\sqrt{n},\dots ,\pm 1/\sqrt{n}\}$ and $n/s\to y>0$ as $n\to \infty$. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix $B_n=𝜃v_n{v}_n^{\ast }+S_n$ is studied where $S_n$ is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or $S_n=M_n$, $𝜃>0$ nonrandom and $v_n$ is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to $v_n$ with the eigenvector associated with the largest eigenvalue of $B_n.$

We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups $\mathrm{\text{O(N)}}$, $\mathrm{\text{U(N)}}$ and $\mathrm{\text{Sp(2N)}}$. For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.