Large deviations of the maximal eigenvalue of random matrices

We present detailed computations of the 'at least finite' terms (three dominant orders) of the free energy in a one-cut matrix model with a hard edge a, in beta-ensembles, with any polynomial potential. beta is a positive number, so not restricted to the standard values beta = 1 (hermitian matrices), beta = 1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This model allows to study the statistic of the maximum eigenvalue of random matrices. We compute the large deviation function to the left of the expected maximum. We specialize our results to the gaussian beta-ensembles and check them numerically. Our method is based on general results and procedures already developed in the literature to solve the Pastur equations (also called "loop equations"). It allows to compute the left tail of the analog of Tracy-Widom laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos corrected and preprint added ; v4 few more numbers adde

Special cases of the orbifold version of Zvonkine's rr-ELSV formula

We prove the orbifold version of Zvonkine's $r$-ELSV formula in two special cases: the case of $r=2$ (complete $3$-cycles) for any genus $g\geq 0$ and the case of any $r\geq 1$ for genus $g=0$

Different Hemodynamic Responses of the Primary Motor Cortex Accompanying Eccentric and Concentric Movements: A Functional NIRS Study

The literature contains limited evidence on how our brains control eccentric movement. A higher activation is expected in the contralateral motor cortex (M1) but consensus has not yet been reached. Therefore, the present study aimed to compare patterns of M1 activation between eccentric and concentric movements. Nine healthy participants performed in a randomized order three sets of five repetitions of eccentric or concentric movement with the dominant elbow flexors over a range of motion of 60◦ at two velocities (30◦/s and 60◦/s). The tests were carried out using a Biodex isokinetic dynamometer with the forearm supported in the horizontal plane. The peak torque values were not significantly different between concentric and eccentric movements (p = 0.42). Hemodynamic responses of the contralateral and ipsilateral M1 were measured with a near-infrared spectroscopy system (Oxymon MkIII, Artinis). A higher contralateral M1 activity was found during eccentric movements (p = 0.04, η2 = 0.47) and at the velocity of 30◦/s (p = 0.039, η2 = 0.48). These preliminary findings indicate a specific control mechanism in the contralateral M1 to produce eccentric muscle actions at the angular velocities investigated, although the role of other brain areas in the motor control network cannot be excluded

Spectral density asymptotics for Gaussian and Laguerre β\beta-ensembles in the exponentially small region

The first two terms in the large $N$ asymptotic expansion of the $\beta$ moment of the characteristic polynomial for the Gaussian and Laguerre $\beta$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.Comment: 19 page

Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation

Using pole decompositions as starting points, the one parameter (-1 =< c < 1) nonlocal and nonlinear Zhdanov-Trubnikov (ZT) equation for the steady shapes of premixed gaseous flames is studied in the large-wrinkle limit. The singular integral equations for pole densities are closely related to those satisfied by the spectral density in the O(n) matrix model, with n = -2(1 + c)/(1 - c). They can be solved via the introduction of complex resolvents and the use of complex analysis. We retrieve results obtained recently for -1 =< c =< 0, and we explain and cure their pathologies when they are continued naively to 0 < c < 1. Moreover, for any -1 =< c < 1, we derive closed-form expressions for the shapes of steady isolated flame crests, and then bicoalesced periodic fronts. These theoretical results fully agree with numerical resolutions. Open problems are evoked.Comment: v2: 29 pages, 6 figures, some typos correcte

Asymptotic forms for hard and soft edge general β\beta conditional gap probabilities

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix $\beta$-ensembles. The conditioning is that there are $n$ eigenvalues in the gap, with $n \ll |t|$, $t$ denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consistent with known asymptotic expansions in the case $n=0$. With this modification made for general $n$, the derived expansions - which are for the logarithm of the gap probabilities - are conjectured to be correct up to and including terms O$(\log|t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating $\beta$ to $4/\beta$.Comment: Replaces v2 which contains typographical errors arising from a previous unpublished draf

Supersymmetric Vacua in Random Supergravity

We determine the spectrum of scalar masses in a supersymmetric vacuum of a general N=1 supergravity theory, with the Kahler potential and superpotential taken to be random functions of N complex scalar fields. We derive a random matrix model for the Hessian matrix and compute the eigenvalue spectrum. Tachyons consistent with the Breitenlohner-Freedman bound are generically present, and although these tachyons cannot destabilize the supersymmetric vacuum, they do influence the likelihood of the existence of an `uplift' to a metastable vacuum with positive cosmological constant. We show that the probability that a supersymmetric AdS vacuum has no tachyons is formally equivalent to the probability of a large fluctuation of the smallest eigenvalue of a certain real Wishart matrix. For normally-distributed matrix entries and any N, this probability is given exactly by P = exp(-2N^2|W|^2/m_{susy}^2), with W denoting the superpotential and m_{susy} the supersymmetric mass scale; for more general distributions of the entries, our result is accurate when N >> 1. We conclude that for |W| \gtrsim m_{susy}/N, tachyonic instabilities are ubiquitous in configurations obtained by uplifting supersymmetric vacua.Comment: 26 pages, 6 figure

From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators

In this letter,we present our conjecture on the connection between the Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the $GL(\infty)$ group element. An important feature of this group element is its simplicity: this is a group element of the Virasoro subalgebra of $gl(\infty)$. If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich--Witten tau-function.Comment: 13 page