How many zeros of a random polynomial are real?

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$ projected onto the surface of the unit sphere, divided by $\pi$. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.Comment: 37 page

On the spectra of Gaussian matrices

AbstractWe give a simple characterization of the moduli of the eigenvalues of a complex Gaussian matrix in terms of x2 distributions. We also show that the spectral radius of a k × k complex Gaussian matrix is stochastically smaller than the norm of a k × (k + 1) real Gaussian matrix

The Ginibre ensemble and Gaussian analytic functions

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to P\'olya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.Comment: 23 pages, 1 figur

BIOLOGICALLY-INFORMED COMPUTATIONAL MODELS OF HARMONIC SOUND DETECTION AND IDENTIFICATION

Harmonic sounds or harmonic components of sounds are often fused into a single percept by the auditory system. Although the exact neural mechanisms for harmonic sensitivity remain unclear, it arises presumably in the auditory cortex because subcortical neurons typically prefer only a single frequency. Pitch sensitive units and harmonic template units found in awake marmoset auditory cortex are sensitive to temporal and spectral periodicity, respectively. This thesis is a study of possible computational mechanisms underlying cortical harmonic selectivity. To examine whether harmonic selectivity is related to statistical regularities of natural sounds, simulated auditory nerve responses to natural sounds were used in principal component analysis in comparison with independent component analysis, which yielded harmonic-sensitive model units with similar population distribution as real cortical neurons in terms of harmonic selectivity metrics. This result suggests that the variability of cortical harmonic selectivity may provide an efficient population representation of natural sounds. Several network models of spectral selectivity mechanisms are investigated. As a side study, adding synaptic depletion to an integrate-and-fire model could explain the observed modulation-sensitive units, which are related to pitch-sensitive units but cannot account for precise temporal regularity. When a feed-forward network is trained to detect harmonics, the result is always a sieve, which is excited by integer multiples of the fundamental frequency and inhibited by half-integer multiples. The sieve persists over a wide variety of conditions including changing evaluation criteria, incorporating Dale’s principle, and adding a hidden layer. A recurrent network trained by Hebbian learning produces harmonic-selective by a novel dynamical mechanism that could be explained by a Lyapunov function which favors inputs that match the learned frequency correlations. These model neurons have sieve-like weights like the harmonic template units when probed by random harmonic stimuli, despite there being no sieve pattern anywhere in the network’s weights. Online stimulus design has the potential to facilitate future experiments on nonlinear sensory neurons. We accelerated the sound-from-texture algorithm to enable online adaptive experimental design to maximize the activities of sparsely responding cortical units. We calculated the optimal stimuli for harmonic-selective units and investigated model-based information-theoretic method for stimulus optimization

Harmonic oscillations and their switching in elliptical optical waveguide arrays

We have studied harmonic oscillations in an elliptical optical waveguide array in which the coupling between neighboring waveguides is varied in accord with a Kac matrix so that the propagation constant eigenvalues can take equally spaced values. As a result, long-living Bloch oscillation (BO) and dipole oscillation (DO) are obtained when a linear gradient in the propagation constant is applied. Moreover, we achieve a switching from DO to BO or vice versa by ramping up the gradient profile. The various optical oscillations as well as their switching are investigated by field evolution analysis and confirmed by Hamiltonian optics. The equally spaced eigenvalues in the propagation constant allow viable applications in transmitting images, switching and routing of optical signals.Comment: 14 pages, 5 figure

On the number of minima of a random polynomial

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain the bound O(exp(-beta n^2 + (n/2) log (d-1))) (beta is a positive constant independent on n and d) for the number of minima of such a polynomial. This proves that most normal random polynomials of fixed degree have only saddle points. Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions.Comment: 22 pages. We learned since the first version that the probability that a matrix in GOE(n) is positive definite is known. This follows from the theory of large deviations (reference in the paper). Therefore, we can now state a precise upper bound (Theorem 2) for the number of minima of a random polynomial, instead of a bound depending on that probabilit

Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.Comment: 37 pages, 2 figures, updated proof

The Dirichlet Markov Ensemble

We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if \bM is such a random matrix, then the empirical distribution built from the singular values of\sqrt{n} \bM tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of \sqrt{n} \bM tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of \bM is of order $1-1/\sqrt{n}$ when $n$ is large.Comment: Improved version. Accepted for publication in JMV

Correlations between zeros of a random polynomial

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval $(-1,1)$ are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random polynomial.Comment: 31 pages, 2 figures; a revised version of the J. Stat. Phys. pape