6,346 research outputs found
When the mean is not enough: Calculating fixation time distributions in birth-death processes
Studies of fixation dynamics in Markov processes predominantly focus on the
mean time to absorption. This may be inadequate if the distribution is broad
and skewed. We compute the distribution of fixation times in one-step
birth-death processes with two absorbing states. These are expressed in terms
of the spectrum of the process, and we provide different representations as
forward-only processes in eigenspace. These allow efficient sampling of
fixation time distributions. As an application we study evolutionary game
dynamics, where invading mutants can reach fixation or go extinct. We also
highlight the median fixation time as a possible analog of mixing times in
systems with small mutation rates and no absorbing states, whereas the mean
fixation time has no such interpretation.Comment: Published in PRE. 14 pages, 6 figure
Localization transition, Lifschitz tails and rare-region effects in network models
Effects of heterogeneity in the suspected-infected-susceptible model on
networks are investigated using quenched mean-field theory. The emergence of
localization is described by the distributions of the inverse participation
ratio and compared with the rare-region effects appearing in simulations and in
the Lifschitz tails. The latter, in the linear approximation, is related to the
spectral density of the Laplacian matrix and to the time dependent order
parameter. I show that these approximations indicate correctly Griffiths Phases
both on regular one-dimensional lattices and on small world networks exhibiting
purely topological disorder. I discuss the localization transition that occurs
on scale-free networks at degree exponent.Comment: 9 pages, 9 figures, accepted version in PR
Non-diagonalizable and non-divergent susceptibility tensor in the Hamiltonian mean-field model with asymmetric momentum distributions
We investigate response to an external magnetic field in the Hamiltonian
mean-field model, which is a paradigmatic toy model of a ferromagnetic body and
consists of plane rotators like the XY spins. Due to long-range interactions,
the external field drives the system to a long-lasting quasistationary state
before reaching thermal equilibrium, and the susceptibility tensor obtained in
the quasista- tionary state is predicted by a linear response theory based on
the Vlasov equation. For spatially homogeneous stable states, whose momentum
distributions are asymmetric with zero-means, the theory reveals that the
susceptibility tensor for an asymptotically constant external field is neither
symmetric nor diagonalizable, and the predicted states are not stationary
accordingly. Moreover, the tensor has no divergence even at the stability
threshold. These theoretical findings are confirmed by direct numerical
simulations of the Vlasov equation for the skew-normal distribution functions.Comment: 10 pages, 8 figure
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