Critical percolation of free product of groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups $G_1,G_2,...,G_n$. For finite groups these equations are polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that $p_c$ for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with the standard generators) is $.5199...$, the unique root of the polynomial $2p^5-6p^4+2p^3+4p^2-1$ in the interval $(0,1)$. In the case when groups $G_i$ can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of $G_1*G_2*...*G_n$ and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, $p_{\mathrm{exp}}$ of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors

Complex networks: new trends for the analysis of brain connectivity

Today, the human brain can be studied as a whole. Electroencephalography, magnetoencephalography, or functional magnetic resonance imaging techniques provide functional connectivity patterns between different brain areas, and during different pathological and cognitive neuro-dynamical states. In this Tutorial we review novel complex networks approaches to unveil how brain networks can efficiently manage local processing and global integration for the transfer of information, while being at the same time capable of adapting to satisfy changing neural demands.Comment: Tutorial paper to appear in the Int. J. Bif. Chao

Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in $\R^d$ is supported on a set of Hausdorff dimension strictly less than $d$ \cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of $\Z ^d$, $d\geq 2$. By refining the argument, we prove that for all \b>0 there exists \rho (d,\b)N(d,\b), any $x \in \Z^d$, and any $A\subset \{1,..., n\}^d$ | \{y\in\Z^d\colon \nu_{A,x}(y) \geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where $\nu_{A,x} (y)$ denotes the probability that $y$ is the first entrance point of the simple random walk starting at $x$ into $A$. Furthermore, $\rho$ must converge to $d$ as \b \to \infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne

Fast computation by block permanents of cumulative distribution functions of order statistics from several populations

The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution function of the populations. The computational cost of the formula in the case of two populations is still exponential in the worst case, but it is a dramatic improvement compared to the general formula by Bapat and Beg. In the case when only the joint distribution function of a subset of the order statistics of fixed size is needed, the complexity is polynomial, for the case of two populations.Comment: 21 pages, 3 figure

Finding Exogenous Variables in Data with Many More Variables than Observations

Many statistical methods have been proposed to estimate causal models in classical situations with fewer variables than observations (p<n, p: the number of variables and n: the number of observations). However, modern datasets including gene expression data need high-dimensional causal modeling in challenging situations with orders of magnitude more variables than observations (p>>n). In this paper, we propose a method to find exogenous variables in a linear non-Gaussian causal model, which requires much smaller sample sizes than conventional methods and works even when p>>n. The key idea is to identify which variables are exogenous based on non-Gaussianity instead of estimating the entire structure of the model. Exogenous variables work as triggers that activate a causal chain in the model, and their identification leads to more efficient experimental designs and better understanding of the causal mechanism. We present experiments with artificial data and real-world gene expression data to evaluate the method.Comment: A revised version of this was published in Proc. ICANN201

A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation

We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^\epsilon$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\omega) = K(p) - V(x/\epsilon,\omega)$ in dimension $d \geq 2$. It is known that homogenization occurs as $\epsilon \to 0$, but little is known about the statistical fluctuations of $w^\epsilon$. Our main result shows that the variance of the solution $w^\epsilon$ is bounded by $O(\epsilon/|\log \epsilon|)$. The proof relies on a modified Poincar\'e inequality of Talagrand

Generative rules of Drosophila locomotor behavior as a candidate homology across phyla

The discovery of shared behavioral processes across phyla is a significant step in the establishment of a comparative study of behavior. We use immobility as an origin and reference for the measurement of fly locomotor behavior; speed, walking direction and trunk orientation as the degrees of freedom shaping this behavior; and cocaine as the parameter inducing progressive transitions in and out of immobility. We characterize and quantify the generative rules that shape Drosophila locomotor behavior, bringing about a gradual buildup of kinematic degrees of freedom during the transition from immobility to normal behavior, and the opposite narrowing down into immobility. Transitions into immobility unfold via sequential enhancement and then elimination of translation, curvature and finally rotation. Transitions out of immobility unfold by progressive addition of these degrees of freedom in the opposite order. The same generative rules have been found in vertebrate locomotor behavior in several contexts (pharmacological manipulations, ontogeny, social interactions) involving transitions in-and-out of immobility. Recent claims for deep homology between arthropod central complex and vertebrate basal ganglia provide an opportunity to examine whether the rules we report also share common descent. Our approach prompts the discovery of behavioral homologies, contributing to the elusive problem of behavioral evolution

The Angular Interval between the Direction of Progression and Body Orientation in Normal, Alcohol- and Cocaine Treated Fruit Flies

In this study we characterize the coordination between the direction a fruit-fly walks and the direction it faces, as well as offer a methodology for isolating and validating key variables with which we phenotype fly locomotor behavior. Our fundamental finding is that the angular interval between the direction a fly walks and the direction it faces is actively managed in intact animals and modulated in a patterned way with drugs. This interval is small in intact flies, larger with alcohol and much larger with cocaine. The dynamics of this interval generates six coordinative modes that flow smoothly into each other. Under alcohol and much more so under cocaine, straight path modes dwindle and modes involving rotation proliferate. To obtain these results we perform high content analysis of video-tracked open field locomotor behavior. Presently there is a gap between the quality of descriptions of insect behaviors that unfold in circumscribed situations, and descriptions that unfold in extended time and space. While the first describe the coordination between low-level kinematic variables, the second quantify cumulative measures and subjectively defined behavior patterns. Here we reduce this gap by phenotyping extended locomotor behavior in terms of the coordination between low-level kinematic variables, which we quantify, combining into a single field two disparate fields, that of high content phenotyping and that of locomotor coordination. This will allow the study of the genes/brain/locomotor coordination interface in genetically engineered and pharmacologically manipulated animal models of human diseases. © 2013 Gakamsky et al

Positive temperature versions of two theorems on first-passage percolation

The estimates on the fluctuations of first-passsage percolation due to Talagrand (a tail bound) and Benjamini--Kalai--Schramm (a sublinear variance bound) are transcribed into the positive-temperature setting of random Schroedinger operators.Comment: 15 pp; to appear in GAFA Seminar Note