Dyson's Brownian motions, intertwining and interlacing

A reflected Brownian motion in the Gelfand-Tsetlin cone is used to construct Dyson's process of non-colliding Brownian motions. The key step of the construction is to consider two interlaced families of Brownian paths with paths belonging to the second family reflected off paths belonging to the first. Such families of paths are known to arise in the Arratia flow of coalescing Brownian motions. A determinantal formula for the distribution of coalescing Brownian motions is presented.

[1]  CONTINUUM-SITES STEPPING-STONE MODELS, COALESCING EXCHANGEABLE PARTITIONS AND RANDOM TREES , 1998, math/9811066.

[2]  P. Bougerol,et al.  Paths in Weyl chambers and random matrices , 2002 .

[3]  K. Ravishankar,et al.  The Brownian Web: Characterization and Convergence , 2004 .

[4]  A Note on Reflecting Brownian Motions , 2002 .

[5]  Littelmann paths and Brownian paths , 2004, math/0403171.

[6]  Gunter M. Schutz.  Exact Solution of the Master Equation for the Asymmetric Exclusion Process , 1997 .

[7]  Reflection and coalescence between independent one-dimensional Brownian paths , 2000 .

[8]  Balls-In-Boxes Duality for Coalescing Random Walks and Coalescing Brownian Motions , 2004, math/0406336.

[9]  A note on representations of eigenvalues of classical Gaussian matrices , 2003 .

[10]  Janko Gravner,et al.  Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models , 2000 .

[11]  W. Koenig.  Orthogonal polynomial ensembles in probability theory , 2004, math/0403090.

[12]  Marc Yor,et al.  A Representation for Non-Colliding Random Walks , 2002 .

[13]  Marc Yor,et al.  Beta-gamma random variables and intertwining relations between certain Markov processes , 1998 .

[14]  K. Johansson.  Shape Fluctuations and Random Matrices , 1999, math/9903134.

[15]  G. Schütz.  Exact solution of the master equation for the asymmetric exclusion process , 1997 .

[16]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[17]  Wendelin Werner,et al.  The true self-repelling motion , 1998 .

[18]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[19]  Reflected planar brownian motions, intertwining relations and crossing probabilities , 2003, math/0302250.

[20]  R. DeBlassie,et al.  The Adjoint Process of Killed Reflected Brownian Motion in a Cone and Applications , 1999 .

[21]  Daniel ben-Avraham.  Complete Exact Solution of Diffusion-Limited Coalescence, A + A → A , 1998 .

[22]  Ruth J. Williams,et al.  Existence and Uniqueness of Semimartingale Reflecting Brownian Motions in Convex Polyhedrons , 1996 .

[23]  R. Rajesh,et al.  Communications in Mathematical Physics Multi-Scaling of the n-Point Density Function for Coalescing Brownian Motions , 2006 .

[24]  Neil O'Connell,et al.  A path-transformation for random walks and the Robinson-Schensted correspondence , 2002, math/0203177.

[25]  F. Dyson.  A Brownian‐Motion Model for the Eigenvalues of a Random Matrix , 1962 .

[26]  Ruth J. Williams,et al.  Multidimensional Reflected Brownian Motions Having Exponential Stationary Distributions , 1987 .

[27]  yuliy baryshnikov.  GUEs and queues , 1999 .