Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

Abstract

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.

Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence

“…In fact, for the 1-matrix model with polynomial potential, this can be proved a posteriori from the asymptotics of M. Bertola [6,7]. But for more general cases, it is only a conjecture, for instance for the 2-matrix model.…”

Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

“…The theory quickly developed and found applications into such fields as the Riemann-Hilbert approach to strong asymptotics, random matrix theory [10,4,5,6,3,2] and in the study of dualities between supersymmetric gauge theories and string models [18,7,11,12,15]. This paper is devoted to a detailed analysis of the phase structure and phase transitions of the asymptotic (in the limit n → ∞) zero density of monic orthogonal polynomials P n (z) = z n + · · · Γ P n (z)z k e −nW (z) dz = 0, k = 0, .…”

Boutroux curves with external field: equilibrium measures without a variational problem

“…Therefore the zero level set X of h(x) is intrinsically well defined and [1] • the set X consists of a finite union of Jordan arcs, some extending to ∞; • all branch-points α j belong to X; • the set X is topologically a forest of trees, 7 namely it does not contain any loop; 6 The integral is defined up to an overall sign but does not depend on the choice of the simple root α 1 because of the Boutroux condition. 7 Here a "forest" means that there may be many connected components each of which is a tree (= does not contain loops).…”

“…This re-formulation is contained in the notion of Boutroux curve and admissibility introduced in [1] and recalled in the next section.…”