The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

120

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t * the smallest paths hit the hard edge and from then on are stuck to it. For t = t * we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

“…Finally, the matching condition (8.5) in condition (4) of the RH problem for Q is satisfied by results of [41]. We have thus established the following.…”

Section: Casementioning

confidence: 69%

Section: Casementioning

confidence: 99%

Section: Casementioning

confidence: 99%

Section: P Is Bounded Asmentioning

confidence: 99%

Section: Casementioning

confidence: 99%

See 3 more Smart Citations

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

120

Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

Duits

2008

Comm Pure Appl Math

133

The eigenvalue statistics of a pair .M 1 ; M 2 / of n n Hermitian matrices taken randomly with respect to the measurecan be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 4 matrix-valued RiemannHilbert problem characterizing one of the families of biorthogonal polynomials in the special case W .y/ D y 4 =4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M 1 (when averaged over M 2 ) in the global and local regime as n ! 1 in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.

Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Zhang

2011

Comm Pure Appl Math

129

We study a model of n one-dimensional non-intersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time-space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new familiy of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new RiemannHilbert problem of size 4 × 4. We prove solvability of the Riemann-Hilbert problem and establish a remarkable connection with the Hastings-McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the non-intersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class.