When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on `bead configurations' on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel.
We propose Mecke-Palm formulas for multiple integrals with respect to a Poisson random measure interlaced with its intensity measure. We apply such formulas to multiple mixed Lévy systems of Lévy processes and obtain moment formulas for products of interlaced multiple Poisson integrals.
We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac1n$, and examine the asymptotic behaviour, as $n \to \infty$, under weak asymptotic assumptions for the (rescaled) particles on row $n$: The empirical d…
We study in some generality intertwinings between $h$-transforms of Karlin-McGregor semigroups associated with one dimensional diffusion processes and those of their Siegmund duals. We obtain couplings so that the corresponding processes are interlaced and furthermore give formulae in terms of block determinants for the transition densities of these coupled processes. This allows us to build diffusion processes in the space of Gelfand-Tsetlin patterns so that the evolution of each level is Markovian. We show how known examples naturally fit into this framework and construct new processes relat…
Cyclic structure and dynamics are of great interest in both the fields of stochastic processes and nonequilibrium statistical physics. In this paper, we find a new symmetry of the Brownian motion named as the quasi-time-reversal invariance. It turns out that such an invariance of the Brownian motion is the key to prove the cycle symmetry for diffusion processes on the circle, which says that the distributions of the forming times of the forward and backward cycles, given that the corresponding cycle is formed earlier than the other, are exactly the same. With the aid of the cycle symmetry, we …
The purpose of this paper is to investigate the long time behaviour for a self-interacting diffusion and a self-interacting velocity jump process. While the diffusion case has already been studied for some particular potential function, the second one, which belongs to the family of piecewise deterministic processes, is new. Depending on the underlying potential function's shape, we prove either the almost sure convergence or the recurrence for a natural extended process given by a change a variable.
This paper defines the notion of generators for a class of decreasing radial Loewner chains which are only continuous with respect to time. For this purpose, "Loewner's integral equation" which generalizes Loewner's differential equation is defined and analyzed. The definition of generators is motivated by the Lévy-Khintchine representation for additive processes on the unit circle. Actually, we can and do introduce a homeomorphism between the above class of Loewner chains and the set of the distributions of increments of additive processes equipped with suitable topologies. On the other hand,…
Two simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile, delay-tolerant networks. We consider two independent walkers moving with constant speed on either the discrete or continuous circle, and changing directions at independent geometric (respectively, exponential) times. One of the walkers carries a message that wishes to travel as far and as fast as possible in the clockwise direction. The message stays with its current carrier unless the two walkers meet, the carrier is moving counter-clockw…
We consider consistent diffusion dynamics, leaving the celebrated Hua-Pickrell measures, depending on a complex parameter $s$, invariant. These, give rise to Feller-Markov processes on the infinite dimensional boundary $Ω$ of the "graph of spectra", the continuum analogue of the Gelfand-Tsetlin graph, via the method of intertwiners of Borodin and Olshanski. In the particular case of $s=0$, this stochastic process is closely related to the $\mathsf{Sine_2}$ point process on $\mathbb{R}$ that describes the spectrum in the bulk of large random matrices. Equivalently, these coherent dynamics are a…
Consider the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measures in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices…
Tacnode GUE-minor Processes and Double Aztec Diamonds
我们研究随机多米诺骨牌平铺的双阿兹特克钻石，一个地区由两个重叠的阿兹特克钻石。随机平铺产生两个离散的决定点过程，称为K-和L-粒子过程。Adler，Johansson and van Moerbeke（2011）推导了K粒子的相关核，他们用它来研究水平和垂直多米诺骨牌的不同权重K粒子的极限过程。让双阿兹特克钻石和重叠的大小趋于无穷大，使得两个北极椭圆刚好接触;然后它们表明，在切点附近的K粒子的波动趋向于交点过程。本文在保证重叠有限的条件下，研究了当水平多米诺骨牌和垂直多米诺骨牌的权值相等或渐近相等时，随着Double Aztec菱形的生长，L粒子在重叠中的极限点过程.在这种情况下，两个极限北极圈在重叠处相切，L粒子在切点附近的行为可以看作是两个碰撞的GUE小过程，我们称之为Tacnode GUE小过程。作为L-粒子的内核的推导的一部分，我们找到了双阿兹特克钻石的二聚体模型版本的逆Kasteleyn矩阵。
