Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

Kuijlaars, Arno B. J.; Martínez–Finkelshtein, Andrei; Wielonsky, Franck

doi:10.1007/s00220-011-1322-x

Cited by 42 publications

(85 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The local path correlations at the origin in Case IV should be the same as those derived in [39], since they both correspond to phase transitions between a soft and a hard edge. The Cases V-VI represent new critical phenomena.…”

Section: Limiting Mean Distributionmentioning

confidence: 87%

“…If t = t * , the local correlations obey the usual scaling limit from the random matrix theory, leading to the sine, Airy, and Bessel kernels [38]. 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%

“…[20,33,36,49]. The case where a > 0 and b = 0 was considered in [38] and [39]. In that situation, all paths, after proper scaling, initially stay away from the hard edge at x = 0, but at a certain critical time t * the lowest paths hit the hard edge and are stuck to it from then on.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Section: Limiting Mean Distributionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

show abstract

“…. , N , [4,14,7,10,9]. The histograms in Figs.1 and 2 show the empirical measures (1.7) for the eigenvalues of matrices L = K † K given by random rectangular matrices K with size 1000 × 300, whose elements are randomly generalized following the probability law (1.23) with (t, a) = (1, 0) and (t, a) = (1, 1), respectively.…”

Section: Resultsmentioning

confidence: 99%

Three-Parametric Marcenko–Pastur Density

Endo

Katori

2020

J Stat Phys

View full text Add to dashboard Cite

The complex Wishart ensemble is the statistical ensemble of M ×N complex random matrices with M ≥ N such that the real and imaginary parts of each element are given by independent standard normal variables. The Marcenko-Pastur (MP) density ρ(x; r), x ≥ 0 describes the distribution for squares of the singular values of the random matrices in this ensemble in the scaling limit N → ∞, M → ∞ with a fixed rectangularity r = N/M ∈ (0, 1]. The dynamical extension of the squared-singular-value distribution is realized by the noncolliding squared Bessel process, and its hydrodynamic limit provides the two-parametric MP density ρ(x; r, t) with time t ≥ 0, whose initial distribution is δ(x). Recently, Blaizot, Nowak, and Warcho l studied the time-dependent complex Wishart ensemble with an external source and introduced the threeparametric MP density ρ(x; r, t, a) by analyzing the hydrodynamic limit of the process starting from δ(x − a), a > 0. In the present paper, we give useful expressions for ρ(x; r, t, a) and perform a systematic study of dynamic critical phenomena observed at the critical time t c (a) = a when r = 1. The universal behavior in the long-term limit t → ∞ is also reported.Keywords Marcenko-Pastur law · Wishart random-matrix ensemble · Wishart process · Random-matrix ensemble with an external source · Hydrodynamic limit · Dynamic critical phenomena

show abstract

“…The critical behavior studied here was however not identified. This was done in the context of non-intersecting squared Bessel paths in [39] where the integral representation of the limiting kernel was derived with Riemann-Hilbert techniques. Finally, this kernel reduces to the so-called symmetric Pearcey kernel identified through the studies of random growth with a wall [40,41].…”

Section: Introductionmentioning

confidence: 99%

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

2014

View full text Add to dashboard Cite

We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.

show abstract

Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

Cited by 42 publications

References 34 publications

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

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

Three-Parametric Marcenko–Pastur Density

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

Contact Info

Product

Resources

About