Limiting dynamics for spherical models of spin glasses at high temperature

We analyze the coupled non-linear integro-differential equations whose solutions is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds

Transition from the annealed to the quenched asymptotics for a random walk on random obstacles

In this work we study a natural transition mechanism describing the passage from a quenched (almost sure) regime to an annealed (in average) one, for a symmetric simple random walk on random obstacles on sites having an identical and independent law. The transition mechanism we study was first proposed in the context of sums of identical independent random exponents by Ben Arous, Bogachev and Molchanov in [Probab. Theory Related Fields 132 (2005) 579--612]. Let $p(x,t)$ be the survival probability at time $t$ of the random walk, starting from site $x$, and let $L(t)$ be some increasing function of time. We show that the empirical average of $p(x,t)$ over a box of side $L(t)$ has different asymptotic behaviors depending on $L(t)$. T here are constants $0<\gamma_1<\gamma_2$ such that if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma>\gamma_1$, a law of large numbers is satisfied and the empirical survival probability decreases like the annealed one; if $L(t)\ge e^{\gamma t^{d/(d+2)}}$, with $\gamma>\gamma_2$, also a central limit theorem is satisfied. If ${L(t)\ll t}$, the averaged survival probability decreases like the quenched survival probability. If $t\ll L(t)$ and $\log L(t)\ll t^{d/(d+2)}$ we obtain an intermediate regime. Furthermore, when the dimension $d=1$ it is possible to describe the fluctuations of the averaged survival probability when $L(t)=e^{\gamma t^{d/(d+2)}}$ with $\gamma<\gamma_2$: it is shown that they are infinitely divisible laws with a L\'{e}vy spectral function which explodes when $x\to0$ as stable laws of characteristic exponent $\alpha<2$. These results show that the quenched and annealed survival probabilities correspond to a low- and high-temperature behavior of a mean-field type phase transition mechanism.Comment: Published at http://dx.doi.org/10.1214/009117905000000404 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Transition asymptotics for reaction-diffusion in random media

We describe a universal transition mechanism characterizing the passage to an annealed behavior and to a regime where the fluctuations about this behavior are Gaussian, for the long time asymptotics of the empirical average of the expected value of the number of random walks which branch and annihilate on ${\mathbb Z}^d$, with stationary random rates. The random walks are independent, continuous time rate $2d\kappa$, simple, symmetric, with $\kappa \ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$, and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates $w(x)=(v_-(x),v_+(x))$ which are i.i.d. We identify a natural way to describe the annealed-Gaussian transition mechanism under mild conditions on the rates. Indeed, we introduce the exponents $F_\theta(t):=\frac{H_1((1+\theta)t)-(1+\theta)H_1(t)}{\theta}$, and assume that $\frac{F_{2\theta}(t)-F_\theta(t)}{\theta\log(\kappa t+e)}\to\infty$ for $|\theta|>0$ small enough, where $H_1(t):=\log$ and $$ denotes the average of the expected value of the number of particles $m(0,t,w)$ at time $t$ and an environment of rates $w$, given that initially there was only one particle at 0. Then the empirical average of $m(x,t,w)$ over a box of side $L(t)$ has different behaviors: if $L(t)\ge e^{\frac{1}{d} F_\epsilon(t)}$ for some $\epsilon >0$ and large enough $t$, a law of large numbers is satisfied; if $L(t)\ge e^{\frac{1}{d} F_\epsilon (2t)}$ for some $\epsilon>0$ and large enough $t$, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative $\epsilon$. Applications to potentials with Weibull, Frechet and double exponential tails are given.Comment: To appear in: Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov, Editors - AMS | CRM, (2007

Slow relaxation, dynamic transitions and extreme value statistics in disordered systems

We show that the dynamics of simple disordered models, like the directed Trap Model and the Random Energy Model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in these models to the extreme value statistics of the associated random energy landscape

Universality of REM-like aging in mean field spin glasses

Aging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model.Comment: 4 page

Principal component-based image segmentation: a new approach to outline in vitro cell colonies

The in vitro clonogenic assay is a technique to study the ability of a cell to form a colony in a culture dish. By optical imaging, dishes with stained colonies can be scanned and assessed digitally. Identification, segmentation and counting of stained colonies play a vital part in high-throughput screening and quantitative assessment of biological assays. Image processing of such pictured/scanned assays can be affected by image/scan acquisition artifacts like background noise and spatially varying illumination, and contaminants in the suspension medium. Although existing approaches tackle these issues, the segmentation quality requires further improvement, particularly on noisy and low contrast images. In this work, we present an objective and versatile machine learning procedure to amend these issues by characterizing, extracting and segmenting inquired colonies using principal component analysis, k-means clustering and a modified watershed segmentation algorithm. The intention is to automatically identify visible colonies through spatial texture assessment and accordingly discriminate them from background in preparation for successive segmentation. The proposed segmentation algorithm yielded a similar quality as manual counting by human observers. High F1 scores (>0.9) and low root-mean-square errors (around 14%) underlined good agreement with ground truth data. Moreover, it outperformed a recent state-of-the-art method. The methodology will be an important tool in future cancer research applications

Flows driven by Banach space-valued rough paths

We show in this note how the machinery of C^1-approximate flows devised in the work "Flows driven by rough paths", and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path, can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force in a simple way.Comment: 8 page

A matrix interpolation between classical and free max operations: I. The univariate case

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.Comment: 14 page