What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution.

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function Z, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for Z to be minimally plausible. Specifically, we present six necessary-but not sufficient-conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken "solutions", unveiling their shortcomings.

Survival in patchy landscapes: the interplay between dispersal, habitat loss and fragmentation.

Habitat loss and fragmentation are important factors determining animal population dynamics and spatial distribution. Such landscape changes can lead to the deleterious impact of a significant drop in the number of species, caused by critically reduced survival rates for organisms. In order to obtain a deeper understanding of the threeway interplay between habitat loss, fragmentation and survival rates, we propose here a spatially explicit multi-scaled movement model of individuals that search for habitat. By considering basic ecological processes, such as predation, starvation (outside the habitat area), and competition, together with dispersal movement as a link among habitat areas, we show that a higher survival rate is achieved in instances with a lower number of patches of larger areas. Our results demonstrate how movement may counterbalance the effects of habitat loss and fragmentation in altered landscapes. In particular, they have important implications for conservation planning and ecosystem management, including the design of specific features of conservation areas in order to enhance landscape connectivity and population viability.

Stochastic optimal foraging: tuning intensive and extensive dynamics in random searches.

Recent theoretical developments had laid down the proper mathematical means to understand how the structural complexity of search patterns may improve foraging efficiency. Under information-deprived scenarios and specific landscape configurations, Lévy walks and flights are known to lead to high search efficiencies. Based on a one-dimensional comparative analysis we show a mechanism by which, at random, a searcher can optimize the encounter with close and distant targets. The mechanism consists of combining an optimal diffusivity (optimally enhanced diffusion) with a minimal diffusion constant. In such a way the search dynamics adequately balances the tension between finding close and distant targets, while, at the same time, shifts the optimal balance towards relatively larger close-to-distant target encounter ratios. We find that introducing a multiscale set of reorientations ensures both a thorough local space exploration without oversampling and a fast spreading dynamics at the large scale. Lévy reorientation patterns account for these properties but other reorientation strategies providing similar statistical signatures can mimic or achieve comparable efficiencies. Hence, the present work unveils general mechanisms underlying efficient random search, beyond the Lévy model. Our results suggest that animals could tune key statistical movement properties (e.g. enhanced diffusivity, minimal diffusion constant) to cope with the very general problem of balancing out intensive and extensive random searching. We believe that theoretical developments to mechanistically understand stochastic search strategies, such as the one here proposed, are crucial to develop an empirically verifiable and comprehensive animal foraging theory.

Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer.

The study of animal foraging behaviour is of practical ecological importance, and exemplifies the wider scientific problem of optimizing search strategies. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical and chemical systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface. This well known finding was followed by similar inferences about the search strategies of deer and bumblebees. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer, microzooplankton, grey seals, spider monkeys and fishing boats. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood and Akaike weights. We apply this to the four original deer and bumblebee data sets, finding that none exhibits evidence of Lévy flights, and that the original graphical approach is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool. Our results question the strength of the empirical evidence for biological Lévy flights.