Non-Equilibrium Dynamics of Dyson’s Model with an Infinite Number of Particles

Katori, Makoto; Tanemura, Hideki

doi:10.1007/s00220-009-0912-3

Cited by 50 publications

(139 citation statements)

References 19 publications

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“…have the same jumps as T has in the respective disks and such that 34) as n → ∞. The construction with Airy functions is well-known in the literature and we do not give details, cf.…”

Section: Local Parametrices Around the Non-critical Endpointsmentioning

confidence: 99%

Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Kuijlaars

Zhang

2011

Comm Pure Appl Math

129

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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.

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“…have the same jumps as T has in the respective disks and such that 34) as n → ∞. The construction with Airy functions is well-known in the literature and we do not give details, cf.…”

Section: Local Parametrices Around the Non-critical Endpointsmentioning

confidence: 99%

Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Kuijlaars

Zhang

2011

Comm Pure Appl Math

129

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“…The important fact is that the obtained interacting particle systems defined in the continuous spatio-temporal plane, which can be called the noncolliding Brownian motion [8], is identified with Dyson's Brownian motion model with β = 2 [9], which was originally introduced as a stochastic process of eigenvalues of an Hermitian-matrix valued Brownian motion in the random matrix theory [10,11]. The notion of correspondence between nonequilibrium particle systems and random matrix theories is very useful [12] and spatio-temporal correlation functions of noncolliding diffusion processes have been determined explicitly not only for the systems with finite numbers of particles but also for the systems with infinite numbers of particles [8,[13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts

Sato

Katori

2011

Phys. Rev. E

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As an image of the many-to-one map of loop-erasing operation L of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW ζ is given by the total weight of all random walks π which are inverse images of ζ, {π : L(π) = ζ}. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce a nonintersecting system of N -tuples of LEBPs in a domain D in the complex plane, where the total weight of nonintersecting LEBPs is given by Fomin's determinant of an N × N matrix whose entries are boundary Poisson kernels in D. We set a sequence of chambers in a planar domain and observe the first passage points at which N Brownian paths (γ 1 , . . . , γ N ) first enter each chamber, under the condition that the loop-erased parts (L(γ 1 ), . . . , L(γ N )) make a system of nonintersecting LEBPs in the domain in the sense of Fomin. We prove that the correlation functions of first passage points of the Brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel. The correlation kernel is of Eynard-Mehta type, which has appeared in two-matrix models and time-dependent matrix models studied in random matrix theory. Conformal covariance of correlation functions is demonstrated.

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“…Note that the behavior of the two differ as the √ s is complex while x is real and because the exponent in the latter has a complex phase. We are however inspired by the analogy and call (23) simply, the Bessoid function.…”

Section: The Characteristic Polynomial At the Critical Pointmentioning

confidence: 99%

“…In the mean time, the theory of non-intersecting Brownian motions or the so called vicious walkers was developed. The subject which originated from the works of de Gennes on fibrous structures [19], and Fisher on wetting and melting [20], was linked to random matrix theory [21][22][23][24], and led to many developments including a physical realization of the statistical properties of Wishart matrices through fluctuations of non-intersecting interfaces in thermal equilibrium [25].…”

Section: Introductionmentioning

confidence: 99%

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

2014

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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.

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Non-Equilibrium Dynamics of Dyson’s Model with an Infinite Number of Particles

Cited by 50 publications

References 19 publications

Critical behavior of nonintersecting Brownian motions at a tacnode

Critical behavior of nonintersecting Brownian motions at a tacnode

Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts

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

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