Construction of a Lax Pair for the E6(1)E_6^{(1)} qq-Painlev\'e System

We construct a Lax pair for the $E^{(1)}_6$ $q$-Painlev\'e system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices - the $q$-linear lattice - through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $E^{(1)}_6$ $q$-Painlev\'e system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations

The Distribution of the first Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble

The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlev\'e V system, and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlev\'e \IIId system and its associated linear isomonodromic system characterise the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter $a$. In particular choosing $a=\pm 1/2$ allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.Comment: 65 pages, 1 eps figure, typos and references correcte

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.Comment: 27 pages, Latex2e plus AMS packages Fix to Eqs. (2.72) and (2.74

Physical Combinatorics and Quasiparticles

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.Comment: 57 pages, in version 2: typos corrected, some sentences clarified, one appendix remove

Comprehensive Approach to Analyzing Rare Genetic Variants

Recent findings suggest that rare variants play an important role in both monogenic and common diseases. Due to their rarity, however, it remains unclear how to appropriately analyze the association between such variants and disease. A common approach entails combining rare variants together based on a priori information and analyzing them as a single group. Here one must make some assumptions about what to aggregate. Instead, we propose two approaches to empirically determine the most efficient grouping of rare variants. The first considers multiple possible groupings using existing information. The second is an agnostic “step-up” approach that determines an optimal grouping of rare variants analytically and does not rely on prior information. To evaluate these approaches, we undertook a simulation study using sequence data from genes in the one-carbon folate metabolic pathway. Our results show that using prior information to group rare variants is advantageous only when information is quite accurate, but the step-up approach works well across a broad range of plausible scenarios. This agnostic approach allows one to efficiently analyze the association between rare variants and disease while avoiding assumptions required by other approaches for grouping such variants

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