David A. Croydon, Tsuyoshi Kato, Makiko Sasada, Satoshi Tsujimoto

2018年6月

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a
cellular automaton that exhibits solitonic behaviour. In this article, we study
the BBS when started from a random two-sided infinite particle configuration.
For such a m...

Ann. Inst. H. Poincare Probab. Statist. 54(4) 1939-1968 2018年 [査読有り]

We establish that if a sequence of spaces equipped with resistance metrics
and measures converge with respect to the Gromov-Hausdorff-vague topology, and
a certain non-explosion condition is satisfied, then the associated stochastic
processes also...

Quenched and annealed heat kernel estimates are established for
Fontes-Isopi-Newman (FIN) processes on spaces equipped with a resistance form.
These results are new even in the case of the one-dimensional FIN diffusion,
and also apply to fractals ...

We discuss the spectral asymptotics of some open subsets of the real line
with random fractal boundary and of a random fractal, the continuum random
tree. In the case of open subsets with random fractal boundary we establish the
existence of the s...

Probab. Theory Related Fields 168 (2017), 269-315 2017年 [査読有り]

We consider the quenched localisation of the Bouchaud trap model on the
positive integers in the case that the trap distribution has a slowly varying
tail at infinity. Our main result is that for each
there exists a slowly...

We introduce and summarise results from the recent paper `Scaling limits of
stochastic processes associated with resistance forms', and also applications
from `Time-changes of stochastic processes associated with resistance forms',
which was writt...

Given a sequence of resistance forms that converges with respect to the
Gromov-Hausdorff-vague topology and satisfies a uniform volume doubling
condition, we show the convergence of corresponding Brownian motions and local
times. As a corollary of...

Annals of Probability 2017, Vol. 45, No. 1, 4-55 2017年 [査読有り]

The first main result of this paper is that the law of the (rescaled)
two-dimensional uniform spanning tree is tight in a space whose elements are
measured, rooted real trees continuously embedded into Euclidean space. Various
properties of the in...

This article describes the quenched localisation behaviour of the Bouchaud
trap model on the integers with regularly varying traps. In particular, it
establishes that for almost every trapping landscape there exist arbitrarily
large times at which...

Annals of Probability 2015, Vol. 43, No. 4, 1594-1642 2015年 [査読有り]

Via a Dirichlet form extension theorem and making full use of two-sided heat
kernel estimates, we establish quenched invariance principles for random walks
in random environments with a boundary. In particular, we prove that the random
walk on a s...

Stoch. Proc. Appl 125(5) (2015), 1980-2009 2015年 [査読有り]

We consider the Bouchaud trap model on the integers in the case that the trap
distribution has a slowly varying tail at infinity. Our main result is a
functional limit theorem for the model under the annealed law, analogous to the
functional limit...

In this article, universal concentration estimates are established for the
local times of random walks on weighted graphs in terms of the resistance
metric. As a particular application of these, a modulus of continuity for local
times is provided ...

David A. Croydon, Alexander Fribergh, Takashi Kumagai

Probab. Theory Related Fields 157(1) 453-507 2013年 [査読有り]

We consider the biased random walk on a critical Galton-Watson tree
conditioned to survive, and confirm that this model with trapping belongs to
the same universality class as certain one-dimensional trapping models with
slowly-varying tails. Inde...

Electronic Journal of Probability 17 (2012), paper no. 3 2012年 [査読有り]

We establish conditions on sequences of graphs which ensure that the mixing
times of the random walks on the graphs in the sequence converge. The main
assumption is that the graphs, associated measures and heat kernels converge in
a suitable Gromo...

We calculate the mean and almost-sure leading order behaviour of the high
frequency asymptotics of the eigenvalue counting function associated with the
natural Dirichlet form on -stable trees, which lead in turn to
short-time heat kernel a...

Electronic Journal of Probability 13 (2008) 1419-1441 2008年 [査読有り]

We establish a variety of properties of the discrete time simple random walk
on a Galton-Watson tree conditioned to survive when the offspring distribution,
say, is in the domain of attraction of a stable law with index
. In pa...

Stochastic Processes and their Applications 118 (2008), no. 5, 730-754 2008年 [査読有り]

We use the random self-similarity of the continuum random tree to show that
it is homeomorphic to a post-critically finite self-similar fractal equipped
with a random self-similar metric. As an application we determine the mean and
almost-sure lea...

In this article, local limit theorems for sequences of simple random walks on
graphs are established. The results formulated are motivated by a variety of
random graph models, and explanations are provided as to how they apply to
supercritical per...

We study the random walk on the range of a simple random walk on
in dimensions . When we establish quenched
and annealed scaling limits for the process , which show that the
intersections of the original si...

Proceedings of the London Mathematical Society 94 (2007), no. 3, 672-694 2007年 [査読有り]

In this article, we consider the problem of estimating the heat kernel on
measure-metric spaces equipped with a resistance form. Such spaces admit a
corresponding resistance metric that reflects the conductivity properties of
the set. In this situ...

Adv. in Appl. Probab. Volume 39, Number 3 (2007), 708-730 2007年 [査読有り]

In this article a collection of random self-similar fractal dendrites is
constructed, and their Hausdorff dimension is calculated. Previous results
determining this quantity for random self-similar structures have relied on
geometrical properties ...

Publications of the Research Institute for Mathematical Sciences 48 (2012), no. 2, 279-338 2012年10月 [査読有り]

A scaling limit for the simple random walk on the largest connected component
of the Erdos-Renyi random graph in the critical window is deduced. The limiting
diffusion is constructed using resistance form techniques, and is shown to
satisfy the sa...

Advances in Applied Probability 42 (2010), no. 2, 528-558 2010年 [査読有り]

Consider a family of random ordered graph trees , where
has vertices. It has previously been established that if the
associated search-depth processes converge to the normalised Brownian excursion
when rescaled appropria...

Probab. Theory Related Fields 157(3) 515-534 2013年 [査読有り]

In this article, a localisation result is proved for the biased random walk
on the range of a simple random walk in high dimensions (d \geq 5). This
demonstrates that, unlike in the supercritical percolation setting, a slowdown
effect occurs as so...

Annals of Probability 2009, Vol. 37, No. 3, 946-978 2009年 [査読有り]

A Brownian spatial tree is defined to be a pair , where
is the rooted real tree naturally associated with a Brownian
excursion and is a random continuous function from into
such...

Ann. Inst. H. Poincare Probab. Statist. 44 (2008), no. 6, 987-1019 2008年 [査読有り]

In this article it is shown that the Brownian motion on the continuum random
tree is the scaling limit of the simple random walks on any family of discrete
-vertex ordered graph trees whose search-depth functions converge to the
Brownian excurs...

Probability Theory and Related Fields 140 (2008), no. 1-2, 207-238 2006年12月 [査読有り]

In this article, we prove global and local (point-wise) volume and heat
kernel bounds for the continuum random tree. We demonstrate that there are
almost-surely logarithmic global fluctuations and log-logarithmic local
fluctuations in the volume o...