Innovation rather than improvement: a solvable high-dimensional model highlights the limitations of scalar fitness

Much of our understanding of ecological and evolutionary mechanisms derives from analysis of low-dimensional models: with few interacting species, or few axes defining "fitness". It is not always clear to what extent the intuition derived from low-dimensional models applies to the complex, high-dimensional reality. For instance, most naturally occurring microbial communities are strikingly diverse, harboring a large number of coexisting species, each of which contributes to shaping the environment of others. Understanding the eco-evolutionary interplay in these systems is an important challenge, and an exciting new domain for statistical physics. Recent work identified a promising new platform for investigating highly diverse ecosystems, based on the classic resource competition model of MacArthur. Here, we describe how the same analytical framework can be used to study evolutionary questions. Our analysis illustrates how, at high dimension, the intuition promoted by a one-dimensional (scalar) notion of fitness can become misleading. Specifically, while the low-dimensional picture emphasizes organism cost or efficiency, we exhibit a regime where cost becomes irrelevant for survival, and link this observation to generic properties of high-dimensional geometry.Comment: 8 pages, 4 figures + Supplementary Materia

Power-law spin correlations in a perturbed honeycomb spin model

We consider spin-$\frac{1}{2}$ model on the honeycomb lattice~\cite{Kitaev06} in presence of a weak magnetic field $h_{\alpha }\ll 1$. Such a perturbation destroys exact integrability of the model in terms of gapless fermions and \textit{static} $Z_{2}$ fluxes. We show that it results in appearance of a long-range tail in the irreducible dynamic spin correlation function: $\left\langle \left\langle s^{z}(t,r)s^{z}(0,0)\right\rangle \right\rangle \propto h_{z}^{2}f(t,r)$, where $f(t,r)\propto \lbrack \max (t,r)]^{-4}$ is proportional to the density polarization function of fermions

Generalized end-product feedback circuit senses high dimensional environmental fluctuations

Understanding computational capabilities of simple biological circuits, such as the regulatory circuits of single-cell organisms, remains an active area of research. Recent theoretical work has shown that a simple regulatory architecture based on end-product inhibition can exhibit predictive behavior by learning fluctuation statistics of one or two environmental parameters. Here we extend this analysis to higher dimensions. We show that as the number of inputs increases, a generalized version of the circuit can learn not only the dominant direction of fluctuations, as shown previously, but also the subdominant fluctuation modes.Comment: 8 pages, 6 figure

Resonances in a single-lead reflection from a disordered medium: σ\sigma-model approach

We develop a general non-perturbative characterisation of universal features of the density $\rho(\Gamma)$ of $S$-matrix poles (resonances) $E_n-i\Gamma_n$ describing waves incident and reflected from a disordered medium via a single $M$-channel waveguide/lead. Explicit expressions for $\rho(\Gamma)$ are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels ($M\sim 1$) the most salient features are tails $\rho(\Gamma)\sim 1/\Gamma$ for narrow resonances reflecting exponential localization and $\rho(\Gamma)\sim 1/\Gamma^2$ for broad resonances reflecting states located in the vicinity of the attached wire. For multimode wires with $M\gg 1$, intermediate asymptotics $\rho(\Gamma)\sim 1/\Gamma^{3/2}$ is shown to emerge reflecting diffusive nature of decay into wide enough contacts.Comment: Article+Supplemental Materia