Adaptive Lévy processes and area-restricted search in human foraging

A considerable amount of research has claimed that animals’ foraging behaviors display movement lengths with power-law distributed tails, characteristic of Lévy flights and Lévy walks. Though these claims have recently come into question, the proposal that many animals forage using Lévy processes nonetheless remains. A Lévy process does not consider when or where resources are encountered, and samples movement lengths independently of past experience. However, Lévy processes too have come into question based on the observation that in patchy resource environments resource-sensitive foraging strategies, like area-restricted search, perform better than Lévy flights yet can still generate heavy-tailed distributions of movement lengths. To investigate these questions further, we tracked humans as they searched for hidden resources in an open-field virtual environment, with either patchy or dispersed resource distributions. Supporting previous research, for both conditions logarithmic binning methods were consistent with Lévy flights and rank-frequency methods–comparing alternative distributions using maximum likelihood methods–showed the strongest support for bounded power-law distributions (truncated Lévy flights). However, goodness-of-fit tests found that even bounded power-law distributions only accurately characterized movement behavior for 4 (out of 32) participants. Moreover, paths in the patchy environment (but not the dispersed environment) showed a transition to intensive search following resource encounters, characteristic of area-restricted search. Transferring paths between environments revealed that paths generated in the patchy environment were adapted to that environment. Our results suggest that though power-law distributions do not accurately reflect human search, Lévy processes may still describe movement in dispersed environments, but not in patchy environments–where search was area-restricted. Furthermore, our results indicate that search strategies cannot be inferred without knowing how organisms respond to resources–as both patched and dispersed conditions led to similar Lévy-like movement distributions

Hide-and-Seek with Directional Sensing

We consider a game played between a hider, who hides a static object in one of several possible positions in a bounded planar region, and a searcher, who wishes to reach the object by querying sensors placed in the plane. The searcher is a mobile agent, and whenever it physically visits a sensor, the sensor returns a random direction, corresponding to a half-plane in which the hidden object is located. We first present a novel search heuristic and characterize bounds on the expected distance covered before reaching the object. Next, we model this game as a large-dimensional zero-sum dynamic game and we apply a recently introduced randomized sampling technique that provides a probabilistic level of security to the hider. We observe that, when the randomized sampling approach is only allowed to select a very small number of samples, the cost of the heuristic is comparable to the security level provided by the randomized procedure. However, as we allow the number of samples to increase, the randomized procedure provides a higher probabilistic security level.Comment: A short version of this paper (without proofs) will be presented at the 18th IFAC World Congress (IFAC 2011), Milan (Italy), August 28-September 2, 201

Phase separation in random cluster models III: circuit regularity

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sites reaches into all parts of the circuit, with maximal distance from one such site to the next being at most logarithmic in n with high probability. The result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations, and, as such, it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in arXiv:1001.1527 and arXiv:1001.1528.Comment: 50 pages, 7 figures. A few typos have been correcte

The Sampling-and-Learning Framework: A Statistical View of Evolutionary Algorithms

Evolutionary algorithms (EAs), a large class of general purpose optimization algorithms inspired from the natural phenomena, are widely used in various industrial optimizations and often show excellent performance. This paper presents an attempt towards revealing their general power from a statistical view of EAs. By summarizing a large range of EAs into the sampling-and-learning framework, we show that the framework directly admits a general analysis on the probable-absolute-approximate (PAA) query complexity. We particularly focus on the framework with the learning subroutine being restricted as a binary classification, which results in the sampling-and-classification (SAC) algorithms. With the help of the learning theory, we obtain a general upper bound on the PAA query complexity of SAC algorithms. We further compare SAC algorithms with the uniform search in different situations. Under the error-target independence condition, we show that SAC algorithms can achieve polynomial speedup to the uniform search, but not super-polynomial speedup. Under the one-side-error condition, we show that super-polynomial speedup can be achieved. This work only touches the surface of the framework. Its power under other conditions is still open

Narrow-escape times for diffusion in microdomains with a particle-surface affinity: Mean-field results

We analyze the mean time t_{app} that a randomly moving particle spends in a bounded domain (sphere) before it escapes through a small window in the domain's boundary. A particle is assumed to diffuse freely in the bulk until it approaches the surface of the domain where it becomes weakly adsorbed, and then wanders diffusively along the boundary for a random time until it desorbs back to the bulk, and etc. Using a mean-field approximation, we define t_{app} analytically as a function of the bulk and surface diffusion coefficients, the mean time it spends in the bulk between two consecutive arrivals to the surface and the mean time it wanders on the surface within a single round of the surface diffusion.Comment: 8 pages, 1 figure, submitted to JC