3,228 research outputs found
Upper large deviations for the maximal flow in first passage percolation
We consider the standard first passage percolation in for
and we denote by the maximal flow through the
cylinder from its bottom to its top. Kesten
proved a law of large numbers for the maximal flow in dimension three: under
some assumptions, converges towards a constant
. We look now at the probability that is
greater than for some , and we show under some
assumptions that this probability decays exponentially fast with the volume of
the cylinder. Moreover, we prove a large deviations principle for the sequence
.Comment: 27 pages, 4 figures; small changes of notation
The spread of a rumor or infection in a moving population
We consider the following interacting particle system: There is a ``gas'' of
particles, each of which performs a continuous-time simple random walk on
, with jump rate . These particles are called -particles
and move independently of each other. They are regarded as individuals who are
ignorant of a rumor or are healthy. We assume that we start the system with
-particles at , and that the ,
are i.i.d., mean- Poisson random variables. In addition, there are
-particles which perform continuous-time simple random walks with jump rate
. We start with a finite number of -particles in the system at time 0.
-particles are interpreted as individuals who have heard a certain rumor or
who are infected. The -particles move independently of each other. The only
interaction is that when a -particle and an -particle coincide, the
latter instantaneously turns into a -particle. We investigate how fast the
rumor, or infection, spreads. Specifically, if
a -particle visits during
and , then we investigate the
asymptotic behavior of . Our principal result states that if
(so that the - and -particles perform the same random walk), then there
exist constants such that almost surely
for all large ,
where . In a further paper we shall use the results
presented here to prove a full ``shape theorem,'' saying that
converges almost surely to a nonrandom set , with the origin as an
interior point, so that the true growth rate for is linear in . If
, then we can only prove the upper bound eventually.Comment: Published at http://dx.doi.org/10.1214/009117905000000413 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains
We consider the following problem in one-dimensional diffusion-limited
aggregation (DLA). At time , we have an "aggregate" consisting of
[with a positive integer]. We also have
particles at , . All these particles perform independent
continuous-time symmetric simple random walks until the first time at
which some particle tries to jump from to . The aggregate is
then increased to the integers in [so that
] and all particles which were at at time are
removed from the system. The problem is to determine how fast grows as a
function of if we start at time 0 with and the i.i.d.
Poisson variables with mean . It is shown that if , then
is of order , in a sense which is made precise. It is conjectured
that will grow linearly in if is large enough.Comment: Published in at http://dx.doi.org/10.1214/07-AOP379 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Diffusion Limited Aggregation on a Cylinder
We consider the DLA process on a cylinder G x N. It is shown that this
process "grows arms", provided that the base graph G has small enough mixing
time. Specifically, if the mixing time of G is at most (log|G|)^(2-\eps), the
time it takes the cluster to reach the m-th layer of the cylinder is at most of
order m |G|/loglog|G|. In particular we get examples of infinite Cayley graphs
of degree 5, for which the DLA cluster on these graphs has arbitrarily small
density.
In addition, we provide an upper bound on the rate at which the "arms" grow.
This bound is valid for a large class of base graphs G, including discrete tori
of dimension at least 3.
It is also shown that for any base graph G, the density of the DLA process on
a G-cylinder is related to the rate at which the arms of the cluster grow. This
implies, that for any vertex transitive G, the density of DLA on a G-cylinder
is bounded by 2/3.Comment: 1 figur
Analytical study of catalytic reactors for hydrazine decomposition Quarterly progress report, 15 Jul. - 14 Oct. 1966
Computer programs used for formulating and solving integral and differential equations in study of catalytic reactors for hydrazine decompositio
Analytical study of catalytic reactors for hydrazine decomposition Quarterly progress report no. 1, 15 Apr. - 14 Jul. 1966
Analytic study of catalytic reactors for hydrazine decompositio
Further evidence on game theory, simulated interaction, and unaided judgement for forecasting decisions in conflicts.
If people in conflicts can more accurately forecast how others will respond, that should help them to make better decisions. Contrary to expert expectations, earlier research found game theorists' forecasts were less accurate than forecasts from simulated interactions using student role players. To assess whether the game theorists had been disadvantaged by the selection of conflicts, I obtained forecasts for three new conflicts (an escalating international confrontation, a takeover battle in the telecommunications industry, and a personal grievance dispute) of types preferred by game theory experts. As before, students were used as role-players, and others provided forecasts using their judgement. When averaged across eight conflicts including five from earlier research, 102 forecasts by 23 game theorists were no more accurate (31% correct predictions) than 357 forecasts by students who used their unaided judgement (32%). Sixty-two percent of 105 simulated-interaction forecasts were accurate, providing an average error reduction of 47% over game-theorist forecasts. Forecasts can sometimes have value without being strictly accurate. Assessing the forecasts using the alternative criterion of usefulness led to the same conclusions about the relative merits of the methods.accuracy, conflict, forecasting, game theory, judgement, methods, role playing, simulated interaction.
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