119 research outputs found
Bouchaud's model exhibits two different aging regimes in dimension one
Let E_i be a collection of i.i.d. exponential random variables. Bouchaud's
model on Z is a Markov chain X(t) whose transition rates are given by
w_{ij}=\nu \exp(-\beta ((1-a)E_i-aE_j)) if i, j are neighbors in Z. We study
the behavior of two correlation functions: P[X(t_w+t)=X(t_w)] and
P[X(t')=X(t_w) \forall t'\in[t_w,t_w+t]]. We prove the (sub)aging behavior of
these functions when \beta >1 and a\in[0,1].Comment: Published at http://dx.doi.org/10.1214/105051605000000124 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Giant vacant component left by a random walk in a random d-regular graph
We study the trajectory of a simple random walk on a d-regular graph with d>2
and locally tree-like structure as the number n of vertices grows. Examples of
such graphs include random d-regular graphs and large girth expanders. For
these graphs, we investigate percolative properties of the set of vertices not
visited by the walk until time un, where u>0 is a fixed positive parameter. We
show that this so-called vacant set exhibits a phase transition in u in the
following sense: there exists an explicitly computable threshold u* such that,
with high probability as n grows, if u<u*, then the largest component of the
vacant set has a volume of order n, and if u>u*, then it has a volume of order
log(n). The critical value u* coincides with the critical intensity of a random
interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular
tree. We also show that the random interlacement model describes the structure
of the vacant set in local neighbourhoods
The Opinion Game: Stock price evolution from microscopic market modelling
We propose a class of Markovian agent based models for the time evolution of
a share price in an interactive market. The models rely on a microscopic
description of a market of buyers and sellers who change their opinion about
the stock value in a stochastic way. The actual price is determined in
realistic way by matching (clearing) offers until no further transactions can
be performed. Some analytic results for a non-interacting model are presented.
We also propose basic interaction mechanisms and show in simulations that these
already reproduce certain particular features of prices in real stock markets.Comment: 14 pages, 5 figure
Aging in two-dimensional Bouchaud's model
Let E_x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on Z^2 is a Markov chain X(t) whose transition rates are given by w_xy=\nu \exp(-\beta E_x) if x, y are neighbours in Z^2. We study the behaviour of two correlation functions: P[X(t_w+t)=X(t_w)] and P[X(t')=X(t_w)\forall t'\in[t_w,t_w+t]]. We prove the (sub)aging behaviour of these functions when \beta >1
- …