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Register a new userPersistence probability of a random polynomial arising from evolutionary game theory
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In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.
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In this paper, using the method proposed by Dembo and Mukherjee [5], we obtain the persistence exponents of random Weyl polynomials in both cases: half nonnegative axis and the whole real axis. Our result is a confirmation to the predictions of Schehr and Majumdar [22].
Many socio-economic and biological processes can be modeled as systems of interacting individuals. The behaviour of such systems can be often described within game-theoretic models. In these lecture notes, we introduce fundamental concepts of evolutionary game theory and review basic properties of deterministic replicator dynamics and stochastic dynamics of finite populations. We discuss stability of equilibria in deterministic dynamics with migration, time-delay, and in stochastic dynamics of well-mixed populations and spatial games with local interactions. We analyze the dependence of the long-run behaviour of a population on various parameters such as the time delay, the noise level, and the size of the population.
Incompatibility of quantum measurements is of fundamental importance in quantum mechanics. It is closely related to many nonclassical phenomena such as Bell nonlocality, quantum uncertainty relations, and quantum steering. We study the necessary and sufficient conditions of quantum compatibility for a given collection of $n$ measurements in $d$-dimensional space. From the compatibility criterion for two-qubit measurements, we compute the incompatibility probability of a pair of independent random measurements. For a pair of unbiased random qubit measurements, we derive that the incompatibility probability is exactly $frac35$. Detailed results are also presented in figures for pairs of general qubit measurements.
Jarzynskis nonequilibrium work relation can be understood as the realization of the (hidden) time-generator reciprocal symmetry satisfied for the conditional probability function. To show this, we introduce the reciprocal process where the classical probability theory is expressed with real wave functions, and derive a mathematical relation using the symmetry. We further discuss that the descriptions by the standard Markov process from an initial equilibrium state are indistinguishable from those by the reciprocal process. Then the Jarzynski relation is obtained from the mathematical relation for the Markov processes described by the Fokker-Planck, Kramers and relativistic Kramers equations.
In this paper we consider an asymptotic question in the theory of the Gaussian Unitary Ensemble of random matrices. In the bulk scaling limit, the probability that there are no eigenvalues in the interval (0,2s) is given by P_s=det(I-K_s), where K_s is the trace-class operator with kernel K_s(x,y)={sin(x-y)}/{pi(x-y)} acting on L^2(0,2s). We are interested particularly in the behavior of P_s as s tends to infinity...
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