26 research outputs found
The time evolution of permutations under random stirring
We consider permutations of obtained by
independent applications of random stirring. In each
step the same marked stirring element is transposed with probability with
any one of the elements. Normalizing by we describe the
asymptotic distribution of the cycle structure of these permutations, for all
, as .Comment: 15 page
Tracy-Widom limit of q-Hahn TASEP
We consider the q-Hahn TASEP which is a three-parameter family of discrete
time interacting particle systems. The particles jump to the right
independently according to a certain q-Binomial distribution with parallel
updates. It is a generalization of the discrete time q-TASEP which is the
q-deformed totally asymmetric simple exclusion process (TASEP) on Z for q in
[0,1). For step initial condition, we prove that the current fluctuation of
q-Hahn TASEP at time t is of order and asymptotically distributed as
the GUE Tracy-Widom distribution. We verify the KPZ scaling theory conjecture
for the q-Hahn TASEP.Comment: 23 pages, 3 figure
The geometry of coalescing random walks, the Brownian web distance and KPZ universality
Coalescing simple random walks in the plane form an infinite tree. A natural
directed distance on this tree is given by the number of jumps between branches
when one is only allowed to move in one direction. The Brownian web distance is
the scale-invariant limit of this directed metric. It is integer-valued and has
scaling exponents 0:1:2 as compared to 1:2:3 in the KPZ world. However, we show
that the shear limit of the Brownian web distance is still given by the Airy
process. We conjecture that our limit theorem can be extended to the full
directed landscape.Comment: 33 pages, 2 figure
The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
A system of non-intersecting squared Bessel processes is considered which all
start from one point and they all return to another point. Under the scaling of
the starting and ending points when the macroscopic boundary of the paths
touches the hard edge, a limiting critical process is described in the
neighbourhood of the touching point which we call the hard edge tacnode
process. We derive its correlation kernel in an explicit new form which
involves Airy type functions and operators that act on the direct sum of
and a finite dimensional space. As the starting points of
the squared Bessel paths are set to 0, a cusp in the boundary appears. The
limiting process is described near the cusp and it is called the hard edge
Pearcey process. We compute its multi-time correlation kernel which extends the
existing formulas for the single-time kernel. Our pre-asymptotic correlation
kernel involves the ratio of two Toeplitz determinants which are rewritten
using a Borodin-Okounkov type formula.Comment: 49 pages, 4 figure
Ages of records in random walks
We consider random walks with continuous and symmetric step distributions. We
prove universal asymptotics for the average proportion of the age of the kth
longest lasting record for k=1,2,... and for the probability that the record of
the kth longest age is broken at step n. Furthermore, we show that the ranked
sequence of proportions of ages converges to the Poisson-Dirichlet
distribution.Comment: 15 pages, 1 figur
Asymptotic fluctuations of geometric q-TASEP, geometric q-PushTASEP and q-PushASEP
We investigate the asymptotic fluctuation of three interacting particle
systems: the geometric q-TASEP, the geometric q-PushTASEP and the q-PushASEP.
We prove that the rescaled particle position converges to the GUE Tracy-Widom
distribution in the homogeneous case. If the jump rates of the first finitely
many particles are perturbed in the first two models, we obtain that the
limiting fluctuations are governed by the Baik-Ben Arous-Peche distribution and
that of the top eigenvalue of finite GUE matrices.Comment: 40 pages, 2 figure
Moments of the superdiffusive elephant random walk with general step distribution
We consider the elephant random walk with general step distribution. We
calculate the first four moments of the limiting distribution of the position
rescaled by in the superdiffusive regime where is the
memory parameter. This extends the results obtained by Bercu.Comment: 11 page
The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
A system of non-intersecting squared Bessel processes is considered which all start from one point and they all return to another point. Under the scaling of the starting and ending points when the macroscopic boundary of the paths touches the hard edge, a limiting critical process is described in the neighbourhood of the touching point which we call the hard edge tacnode process. We derive its corre- lation kernel in an explicit new form which involves Airy type functions and oper- ators that act on the direct sum of L2(R+) and a finite dimensional space. As the starting points of the squared Bessel paths are set to 0, a cusp in the boundary appears. The limiting process is described near the cusp and it is called the hard edge Pearcey process. We compute its multi-time correlation kernel which extends the existing formulas for the single-time kernel. Our pre-asymptotic correlation kernel involves the ratio of two Toeplitz determinants which are rewritten using a Borodin–Okounkov type formula