581 research outputs found
Diffusive and Super-Diffusive Limits for Random Walks and Diffusions with Long Memory
We survey recent results of normal and anomalous diffusion of two types of
random motions with long memory in or . The first
class consists of random walks on in divergence-free random drift
field, modelling the motion of a particle suspended in time-stationary
incompressible turbulent flow. The second class consists of self-repelling
random diffusions, where the diffusing particle is pushed by the negative
gradient of its own occupation time measure towards regions less visited in the
past. We establish normal diffusion (with square-root-of-time scaling and
Gaussian limiting distribution) in three and more dimensions and typically
anomalously fast diffusion in low dimensions (typically, one and two). Results
are quoted from various papers published between 2012-2018, with some hints to
the main ideas of the proofs. No technical details are presented here.Comment: ICM-2018 Probability Section tal
Marginal densities of the "true" self-repelling motion
Let X(t) be the true self-repelling motion (TSRM) constructed by B.T. and
Wendelin Werner in 1998, L(t,x) its occupation time density (local time) and
H(t):=L(t,X(t)) the height of the local time profile at the actual position of
the motion. The joint distribution of (X(t),H(t)) was identified by B.T. in
1995 in somewhat implicit terms. Now we give explicit formulas for the
densities of the marginal distributions of X(t) and H(t). The distribution of
X(t) has a particularly surprising shape: It has a sharp local minimum with
discontinuous derivative at 0. As a consequence we also obtain a precise
version of the large deviation estimate of arXiv:1105.2948v3.Comment: 20 pages, 7 figure
QCD finite T transition -- Comparison between Wilson and staggered results
A quantitative comparison between the finite temperature behaviour of the
staggered and Wilson fermion formulations are performed. The comparison is
based on a physical quantity that is expected to be quite sensitive to the
fermionic features of the action. For that purpose we use the height of the
peak for , where is the quark number susceptibility.Comment: 6 pages. Talk presented at Lattice 200
Cluster growth in the dynamical Erd\H{o}s-R\'{e}nyi process with forest fires
We investigate the growth of clusters within the forest fire model of
R\'{a}th and T\'{o}th [22]. The model is a continuous-time Markov process,
similar to the dynamical Erd\H{o}s-R\'{e}nyi random graph but with the addition
of so-called fires. A vertex may catch fire at any moment and, when it does so,
causes all edges within its connected cluster to burn, meaning that they
instantaneously disappear. Each burned edge may later reappear.
We give a precise description of the process of the size of the cluster
of a tagged vertex, in the limit as the number of vertices in the model tends
to infinity. We show that is an explosive branching process with a
time-inhomogeneous offspring distribution and instantaneous return to on
each explosion. Additionally, we show that the characteristic curves used to
analyse the Smoluchowski-type coagulation equations associated to the model
have a probabilistic interpretation in terms of the process .Comment: 31 page
Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem
We show that a large collection of statistical mechanical systems with
quadratically represented Hamiltonians on the complete graph can be extended to
infinite exchangeable processes. This extends a known result for the
ferromagnetic Curie--Weiss Ising model and includes as well all ferromagnetic
Curie--Weiss Potts and Curie--Weiss Heisenberg models. By de Finetti's theorem,
this is equivalent to showing that these probability measures can be expressed
as averages of product measures. We provide examples showing that
``ferromagnetism'' is not however in itself sufficient and also study in some
detail the Curie--Weiss Ising model with an additional 3-body interaction.
Finally, we study the question of how much the antiferromagnetic Curie--Weiss
Ising model can be extended. In this direction, we obtain sharp asymptotic
results via a solution to a new moment problem. We also obtain a ``formula''
for the extension which is valid in many cases.Comment: Published at http://dx.doi.org/10.1214/009117906000001033 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Improvements of Hungarian Hidden Markov Model-based text-to-speech synthesis
Statistical parametric, especially Hidden Markov Model-based, text-to-speech (TTS) synthesis has received much attention recently. The quality of HMM-based speech synthesis approaches that of the state-of-the-art unit selection systems and possesses numerous favorable features, e.g. small runtime footprint, speaker interpolation, speaker adaptation. This paper presents the improvements of a Hungarian HMM-based speech synthesis system, including speaker dependent and adaptive training, speech synthesis with pulse-noise and mixed excitation. Listening tests and their evaluation are also described
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