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

Higher Order Analogues of Tracy-Widom Distributions via the Lax Method

We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painleve XXXIV hierarchy. These are the higher order analogues of the Tracy-Widom distribution which has k=1. Using known Backlund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painleve II hierarchy.Comment: 24 pages. Improved discussion of Backlund transformations, in addition to other minor improvements in text. Typos corrected. Matches published versio

Right tail expansion of Tracy-Widom beta laws

Using loop equations, we compute the large deviation function of the maximum eigenvalue to the right of the spectrum in the Gaussian beta matrix ensembles, to all orders in 1/N. We then give a physical derivation of the all order asymptotic expansion of the right tail Tracy-Widom beta laws, for all positive beta, by studying the double scaling limit.Comment: 23 page

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

Prefrontal cortex activity and functional organisation in dual-task ocular pursuit is affected by concurrent upper limb movement

Tracking a moving object with the eyes seems like a simple task but involves areas of prefrontal cortex (PFC) associated with attention, working memory and prediction. Increasing the demand on these processes with secondary tasks can affect eye movements and/or perceptual judgments. This is particularly evident in chronic or acute neurological conditions such as Alzheimer’s disease or mild traumatic brain injury. Here, we combined near infrared spectroscopy and video-oculography to examine the effects of concurrent upper limb movement, which provides additional afference and efference that facilitates tracking of a moving object, in a novel dual-task pursuit protocol. We confirmed the expected effects on judgement accuracy in the primary and secondary tasks, as well as a reduction in eye velocity when the moving object was occluded. Although there was limited evidence of oculo-manual facilitation on behavioural measures, performing concurrent upper limb movement did result in lower activity in left medial PFC, as well as a change in PFC network organisation, which was shown by Graph analysis to be locally and globally more efficient. These findings extend upon previous work by showing how PFC is functionally organised to support eye-hand coordination when task demands more closely replicate daily activities

Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case

In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ models. Our approach follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinantal formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts.Comment: 37 pages, 4 figures. Presentation improved, typos corrected. Published in Journal Of Statistical Mechanic

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

A unified fluctuation formula for one-cut β\beta-ensembles of random matrices

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $\beta$-ensemble. As particular cases of the main result we consider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.Comment: 16 pages, 4 figures, 3 tables. Revised version where references have been added and typos correcte

Eccentric cycling involves greater mental demand and cortical activation of the frontoparietal network

Eccentric, compared to concentric exercise, is proposed to involve different neuro-motor processing strategies and a higher level of mental demand. This study compared eccentric and concentric cycling at matched perceived effort and torque for the mental demand and related-cortical activation patterns. Nineteen men (30 ± 6 years) performed four different 5-min cycling conditions at 30 RPM on a semi-recumbent isokinetic cycle ergometer: (1) concentric at a moderate perceived effort (23 on the CR100® scale) without torque feedback; (2) concentric and (3) eccentric at the same average torque produced in the first condition; and (4) eccentric at the same moderate perceived effort than the first concentric condition. The conditions two to four were randomized. After each condition, mental demand was monitored using the NASA Task Load Index scale. Changes in oxy-(O2Hb) and deoxy-(HHb) hemoglobin during exercise were measured over both prefrontal cortices and the right parietal lobe from a 15-probe layout using a continuous-wave NIRS system. Mental demand was significantly higher during eccentric compared to concentric cycling (+52%, p = 0.012) and when the exercise intensity was fixed by the torque rather than the perceived effort (+70%, p < 0.001). For both torque- or perceived effort-matched exercises, O2Hb increased significantly (p < 0.001) in the left and right prefrontal cortices, and right parietal lobe, and HHb decreased in the left, and right, prefrontal cortices during eccentric compared to concentric cycling. This study supports that acute eccentric cycling, compared to concentric cycling, involves a higher mental demand, and frontoparietal network activation