3,954 research outputs found
Beta-gamma tail asymptotics
We compute the tail asymptotics of the product of a beta random variable and
a generalized gamma random variable which are independent and have general
parameters. A special case of these asymptotics were proved and used in a
recent work of Bubeck, Mossel, and R\'acz in order to determine the tail
asymptotics of the maximum degree of the preferential attachment tree. The
proof presented here is simpler and highlights why these asymptotics hold.Comment: 6 page
Optimal control for diffusions on graphs
Starting from a unit mass on a vertex of a graph, we investigate the minimum
number of "\emph{controlled diffusion}" steps needed to transport a constant
mass outside of the ball of radius . In a step of a controlled diffusion
process we may select any vertex with positive mass and topple its mass equally
to its neighbors. Our initial motivation comes from the maximum overhang
question in one dimension, but the more general case arises from optimal mass
transport problems.
On we show that steps are necessary and
sufficient to transport the mass. We also give sharp bounds on the comb graph
and -ary trees. Furthermore, we consider graphs where simple random walk has
positive speed and entropy and which satisfy Shannon's theorem, and show that
the minimum number of controlled diffusion steps is , where is the Avez asymptotic entropy and is the speed
of random walk. As examples, we give precise results on Galton-Watson trees and
the product of trees .Comment: 32 pages, 2 figure
Order statistics of 1/f^{\alpha} signals
Order statistics of periodic, Gaussian noise with 1/f^{\alpha} power spectrum
is investigated. Using simulations and phenomenological arguments, we find
three scaling regimes for the average gap d_k= between the k-th
and (k+1)-st largest values of the signal. The result d_k ~ 1/k known for
independent, identically distributed variables remains valid for 0<\alpha<1.
Nontrivial, \alpha-dependent scaling exponents d_k ~ k^{(\alpha -3)/2} emerge
for 1<\alpha<5 and, finally, \alpha-independent scaling, d_k ~ k is obtained
for \alpha>5. The spectra of average ordered values \epsilon_k= ~
k^{\beta} is also examined. The exponent {\beta} is derived from the gap
scaling as well as by relating \epsilon_k to the density of near extreme
states. Known results for the density of near extreme states combined with
scaling suggest that \beta(\alpha=2)=1/2, \beta(4)=3/2, and beta(infinity)=2
are exact values. We also show that parallels can be drawn between \epsilon_k
and the quantum mechanical spectra of a particle in power-law potentials.Comment: 8 pages, 5 figure
Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction
We introduce and investigate a new model of a finite number of particles
jumping forward on the real line. The jump lengths are independent of
everything, but the jump rate of each particle depends on the relative position
of the particle compared to the center of mass of the system. The rates are
higher for those left behind, and lower for those ahead of the center of mass,
providing an attractive interaction keeping the particles together. We prove
that in the fluid limit, as the number of particles goes to infinity, the
evolution of the system is described by a mean field equation that exhibits
traveling wave solutions. A connection to extreme value statistics is also
provided.Comment: 35 pages, 9 figures. A shortened version appears as arXiv:1108.243
A Smooth Transition from Powerlessness to Absolute Power
We study the phase transition of the coalitional manipulation problem for
generalized scoring rules. Previously it has been shown that, under some
conditions on the distribution of votes, if the number of manipulators is
, where is the number of voters, then the probability that a
random profile is manipulable by the coalition goes to zero as the number of
voters goes to infinity, whereas if the number of manipulators is
, then the probability that a random profile is manipulable
goes to one. Here we consider the critical window, where a coalition has size
, and we show that as goes from zero to infinity, the limiting
probability that a random profile is manipulable goes from zero to one in a
smooth fashion, i.e., there is a smooth phase transition between the two
regimes. This result analytically validates recent empirical results, and
suggests that deciding the coalitional manipulation problem may be of limited
computational hardness in practice.Comment: 22 pages; v2 contains minor changes and corrections; v3 contains
minor changes after comments of reviewer
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