12 research outputs found
Random geometry and the Kardar-Parisi-Zhang universality class
We consider a model of a quenched disordered geometry in which a random metric is defined on R-2, which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius R scales as R-x, with a fluctuation exponent x similar or equal to 1/3, while the lateral spread of the minimizing geodesic between two points at a distance L grows as L-zeta, with wandering exponent value zeta similar or equal to 2/3. Results on related first-passage percolation problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang universality class of surface kinetic roughening, with. and. relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy-Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW-GUE statistics with good accuracy in arrival times.We want to acknowledge very useful discussions with K Takeuchi and S Ferreira. This work has been supported by the Spanish government (MINECO) through grant FIS2012-38866-C05-01. JR-L also acknowledges MINECO grants FIS2012-33642, TOQATA and ERC grant QUAGATUA. TLʼs research and travel was supported in part by NSF PIRE grant OISE-07-30136
Geodesics of Random Riemannian Metrics
We analyze the disordered Riemannian geometry resulting from random
perturbations of the Euclidean metric. We focus on geodesics, the paths traced
out by a particle traveling in this quenched random environment. By taking the
point of the view of the particle, we show that the law of its observed
environment is absolutely continuous with respect to the law of the random
metric, and we provide an explicit form for its Radon-Nikodym derivative. We
use this result to prove a "local Markov property" along an unbounded geodesic,
demonstrating that it eventually encounters any type of geometric phenomenon.
We also develop in this paper some general results on conditional Gaussian
measures. Our Main Theorem states that a geodesic chosen with random initial
conditions (chosen independently of the metric) is almost surely not
minimizing. To demonstrate this, we show that a minimizing geodesic is
guaranteed to eventually pass over a certain "bump surface," which locally has
constant positive curvature. By using Jacobi fields, we show that this is
sufficient to destabilize the minimizing property.Comment: 55 pages. Supplementary material at arXiv:1206.494
Group Incentives and Rational Voting
Our model describes competition between groups driven by the choices of
self-interested voters within groups. Within a Poisson voting environment,
parties observe aggregate support from groups and can allocate prizes or
punishments to them. In a tournament style analysis, the model characterizes
how contingent allocation of prizes based on relative levels of support affects
equilibrium voting behavior. In addition to standard notions of pivotality,
voters influence the distribution of prizes across groups. Such prize
pivotality supports positive voter turnout even in non-competitive electoral
settings. The analysis shows that competition for a prize awarded to the most
supportive group is only stable when two groups actively support a party.
However, competition among groups to avoid punishment is stable in environments
with any number of groups. We conclude by examining implications for endogenous
group formation and how politicians structure the allocation of rewards and
punishments.Comment: 34 pages, 1 figur
Recommended from our members
Geodesics of Random Riemannian Metrics
We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on R^d . We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP
The Mathematics of Evolution: The Price Equation, Natural Selection, and Environmental Change
We extend George Price's evolutionary framework to the measure-theoretic and
quantum cases, decomposing all processes into selective and environmental
components. We further quantify selection and environmental change using
entropy functionals: selective entropy is non-positive, representing biological
negentropy, and environmental entropy is non-negative, representing physical
entropy. We prove four Laws of Natural Selection, showing that selection
consistently acts in a manner to increase selection, but which can be disrupted
by environmental change.Comment: 64 pages, appendix 11 page