The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_beta(N,r)$ consist of $N times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uni
form measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N to infty$. In the unitary case ($beta=2$), we also find an explicit formula for the empirical spectral density for each $N$.
Consider a group word w in n letters. For a compact group G, w induces a map G^n rightarrow G$ and thus a pushforward measure {mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem
2.5 that {mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsens classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {mu}_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These formulas generalize a result of ErdH{o}s and Turan on the probability that two random elements of a finite group commute [ET68]. As another corollary, we give an elementary proof that the dimension of an irreducible complex representation of a finite group divides the order of the group; the only ingredients are Schurs lemma, basic counting, and a divisibility argument.
Alice seeks an information-theoretically secure source of private random data. Unfortunately, she lacks a personal source and must use remote sources controlled by other parties. Alice wants to simulate a coin flip of specified bias $alpha$, as a fun
ction of data she receives from $p$ sources; she seeks privacy from any coalition of $r$ of them. We show: If $p/2 leq r < p$, the bias can be any rational number and nothing else; if $0 < r < p/2$, the bias can be any algebraic number and nothing else. The proof uses projective varieties, convex geometry, and the probabilistic method. Our results improve on those laid out by Yao, who asserts one direction of the $r=1$ case in his seminal paper [Yao82]. We also provide an application to secure multiparty computation.
Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin patterned ensemble
s sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a dial we can turn from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f_m show a visually stunning convergence to the semi-circle as m tends to infinity, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f_m is the product of a Gaussian and a degree 2m-2 polynomial; the formula equals that of the m x m Gaussian Unitary Ensemble (GUE). The proof is by the moments. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.
