16 research outputs found
Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime
We consider a one-dimensional system of Brownian particles that repel each
other through a logarithmic potential. We study two formulations for the system
and the relation between them. The first, Dyson's Brownian motion model, has an
interaction coupling constant determined by the parameter beta > 0. When beta =
1,2 and 4, this model can be regarded as a stochastic realization of the
eigenvalue statistics of Gaussian random matrices. The second system comes from
Dunkl processes, which are defined using differential-difference operators
(Dunkl operators) associated with finite abstract vector sets called root
systems. When the type-A root system is specified, Dunkl processes constitute a
one-parameter system similar to Dyson's model, with the difference that its
particles interchange positions spontaneously. We prove that the type-A Dunkl
processes with parameter k > 0 starting from any symmetric initial
configuration are equivalent to Dyson's model with the parameter beta = 2k. We
focus on the intertwining operators, since they play a central role in the
mathematical theory of Dunkl operators, but their general closed form is not
yet known. Using the equivalence between symmetric Dunkl processes and Dyson's
model, we extract the effect of the intertwining operator of type A on
symmetric polynomials from these processes' transition probability densities.
In the strong coupling limit, the intertwining operator maps all symmetric
polynomials onto a function of the sum of their variables. In this limit,
Dyson's model freezes, and it becomes a deterministic process with a final
configuration proportional to the roots of the Hermite polynomials multiplied
by the square root of the process time, while being independent of the initial
configuration.Comment: LaTeX, 30 pages, 1 figure, 1 table. Corrected for submission to
Journal of Physics
Large deviations for clocks of self-similar processes
The Lamperti correspondence gives a prominent role to two random time
changes: the exponential functional of a L\'evy process drifting to
and its inverse, the clock of the corresponding positive self-similar process.
We describe here asymptotical properties of these clocks in large time,
extending the results of Yor and Zani
Intertwinings for general ÎČ Laguerre and Jacobi processes
We show that, for ÎČâ„1, the semigroups of ÎČ-Laguerre and ÎČ-Jacobi processes of different dimensions are intertwined in analogy to a similar result for ÎČ-Dyson Brownian motion recently obtained in Ramanan and Shkolnikov (Intertwinings of ÎČ-Dyson Brownian motions of different dimensions, 2016. arXiv:1608.01597). These intertwining relations generalize to arbitrary ÎČâ„1 the ones obtained for ÎČ=2 in Assiotis et al. (Interlacing diffusions, 2016. arXiv:1607.07182) between h-transformed KarlinâMcGregor semigroups. Moreover, they form the key step toward constructing a multilevel process in a GelfandâTsetlin pattern leaving certain Gibbs measures invariant. Finally, as a by-product, we obtain a relation between general ÎČ-Jacobi ensembles of different dimensions
Trapped surfaces and symmetries
We prove that strictly stationary spacetimes cannot contain closed trapped
nor marginally trapped surfaces. The result is purely geometric and holds in
arbitrary dimension. Other results concerning the interplay between
(generalized) symmetries and trapped submanifolds are also presented.Comment: 9 pages, no figures. Final corrected version to appear in Class.
Quantum Gra