Random surface growth with a wall and Plancherel measures for <i>O</i> (∞)

Borodin, Alexei; Kuan, Jeffrey

doi:10.1002/cpa.20320

Cited by 57 publications

(133 citation statements)

References 53 publications

(115 reference statements)

Supporting

Mentioning

129

Contrasting

Order By: Relevance

“…This is again a reformulation of the classical branching rules for the representations of Sp(2N ) and SO(N ). There are various formulations of the branching rules, the required form for the symplectic group is best explained in [Kir89] and for the orthogonal groups the lozenge tilings interpretation equivalent to weakly symmetric tilings is described in [BK10]. as N → ∞ for any 0 < x < 1 and any y ∈ R the normalized height functions 1 N H(Nx, Ny) and 1 N H(Nx, Ny) converge to the same deterministic limit function.…”

Section: Restrictions and Lozenge Tilingsmentioning

confidence: 99%

“…Analogues of the Plancherel measure for groups U (∞) and SO(∞) and their asymptotics were studied only much later by Borodin and Kuan [BK08,BK10]. In a recent article [BBO15] Borodin, Olshanski and one of the authors obtained the limit shape theorem for the restrictions of a rich family of characters (representations) of U (∞).…”

Section: U (∞) and Infinite Divisibilitymentioning

confidence: 99%

See 1 more Smart Citation

Representations of classical Lie groups and quantized free convolution

Bufetov

Gorin

2015

Geom. Funct. Anal.

View full text Add to dashboard Cite

Abstract. We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

show abstract

Section: Restrictions and Lozenge Tilingsmentioning

confidence: 99%

Section: U (∞) and Infinite Divisibilitymentioning

confidence: 99%

Representations of classical Lie groups and quantized free convolution

Bufetov

Gorin

2015

Geom. Funct. Anal.

View full text Add to dashboard Cite

show abstract

“…(b) The support of µ 2 is bounded and there exist 0 ≤ p < q such that 10) and µ 2 has a density which is real analytic in the interior of the support and vanishes like a square root at the endpoint q and at p if p > 0.…”

Section: A Vector Equilibrium Problemmentioning

confidence: 99%

“…A new kernel was established near the origin at the critical time t * in [39]. The kernel admits a double integral representation which resembles the Pearcey kernel [8,10,12].…”

Section: Introductionmentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

View full text Add to dashboard Cite

We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T . After proper scaling, the paths fill out a region in the tx-plane. Depending on the value of the product ab the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.

show abstract

“…Their discrete counterparts, the non-intersecting random walks, have important connections with tiling and random growth models, see e.g. [8,12,28,29,42].…”

Section: Introductionmentioning

confidence: 99%

Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Kuijlaars

Zhang

2011

Comm Pure Appl Math

129

View full text Add to dashboard Cite

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.

show abstract

Random surface growth with a wall and Plancherel measures for O (∞)

Cited by 57 publications

References 53 publications

Representations of classical Lie groups and quantized free convolution

Representations of classical Lie groups and quantized free convolution

Non-intersecting squared Bessel paths with one positive starting and ending point

Critical behavior of nonintersecting Brownian motions at a tacnode

Contact Info

Product

Resources

About