Ergodicity and Slowing Down in Glass-Forming Systems with Soft Potentials: No Finite-Temperature Singularities

The aim of this paper is to discuss some basic notions regarding generic glass forming systems composed of particles interacting via soft potentials. Excluding explicitly hard-core interaction we discuss the so called `glass transition' in which super-cooled amorphous state is formed, accompanied with a spectacular slowing down of relaxation to equilibrium, when the temperature is changed over a relatively small interval. Using the classical example of a 50-50 binary liquid of N particles with different interaction length-scales we show that (i) the system remains ergodic at all temperatures. (ii) the number of topologically distinct configurations can be computed, is temperature independent, and is exponential in N. (iii) Any two configurations in phase space can be connected using elementary moves whose number is polynomially bounded in N, showing that the graph of configurations has the `small world' property. (iv) The entropy of the system can be estimated at any temperature (or energy), and there is no Kauzmann crisis at any positive temperature. (v) The mechanism for the super-Arrhenius temperature dependence of the relaxation time is explained, connecting it to an entropic squeeze at the glass transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0Comment: 10 pages, 9 figures, submitted to PR

Average shape of fluctuations for subdiffusive walks

We study the average shape of fluctuations for subdiffusive processes, i.e., processes with uncorrelated increments but where the waiting time distribution has a broad power-law tail. This shape is obtained analytically by means of a fractional diffusion approach. We find that, in contrast with processes where the waiting time between increments has finite variance, the fluctuation shape is no longer a semicircle: it tends to adopt a table-like form as the subdiffusive character of the process increases. The theoretical predictions are compared with numerical simulation results.Comment: 4 pages, 6 figures. Accepted for publication Phys. Rev. E (Replaced for the latest version, in press.) Section II rewritte

Slow movement of a random walk on the range of a random walk in the presence of an external field

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d \geq 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime

Aging in the random energy model

In this letter we announce rigorous results on the phenomenon of aging in the Glauber dynamics of the random energy model and their relation to Bouchaud's 'REM-like' trap model. We show that, below the critical temperature, if we consider a time-scale that diverges with the system size in such a way that equilibrium is almost, but not quite reached on that scale, a suitably defined autocorrelation function has the same asymptotic behaviour than its analog in the trap model.Comment: 4pp, P

From interacting particle systems to random matrices

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of hermitian random matrices, but this analogy does not extend to the case of symmetric matrices. Therefore the connections between growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor corrections in scaling of section 2.

On small-noise equations with degenerate limiting system arising from volatility models

The one-dimensional SDE with non Lipschitz diffusion coefficient $dX_{t} = b(X_{t})dt + \sigma X_{t}^{\gamma} dB_{t}, \ X_{0}=x, \ \gamma<1$ is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and S.~Violante. Marginal density expansions for diffusions and stochastic volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with the rescaled variable $X^{\varepsilon}:=\varepsilon^{1/(1-\gamma)} X$: while allowing to turn a space asymptotic problem into a small-$\varepsilon$ problem with fixed terminal point, the process $X^{\varepsilon}$ satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise $\varepsilon dB$). We prove a pathwise large deviation principle for the process $X^{\varepsilon}$ as $\varepsilon \to 0$. As it will become clear, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell--Freidlin theory. As for applications, the $\varepsilon$-scaling allows to derive exact log-asymptotics for path functionals of the process: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving such an SDE as a component.Comment: 21 pages, 1 figur

On universality of local edge regime for the deformed Gaussian Unitary Ensemble

We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$.Comment: 25 pages, 2 figure

Fluctuations for the Ginzburg-Landau ∇ϕ\nabla \phi Interface Model on a Bounded Domain

We study the massless field on $D_n = D \cap \tfrac{1}{n} \Z^2$, where $D \subseteq \R^2$ is a bounded domain with smooth boundary, with Hamiltonian \CH(h) = \sum_{x \sim y} \CV(h(x) - h(y)). The interaction \CV is assumed to be symmetric and uniformly convex. This is a general model for a $(2+1)$-dimensional effective interface where $h$ represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: $h(x) = n x \cdot u + f(x)$ for $x \in \partial D_n$, $u \in \R^2$, and $f \colon \R^2 \to \R$ continuous. We prove that the fluctuations of linear functionals of $h(x)$ about the tilt converge in the limit to a Gaussian free field on $D$, the standard Gaussian with respect to the weighted Dirichlet inner product $(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i$ for some explicit $\beta = \beta(u)$. In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of $h$ are asymptotically described by $SLE(4)$, a conformally invariant random curve.Comment: 58 page

A critical inventory and associated chronology of the Middle Stone Age and Later Stone Age in Northwest Africa

The study of human evolution and cultural patterns relies on empirical evidence provided by the archaeological record. Accessing dependable archaeological data from scholarly publications can often be challenging due to the variability in site documentation and the diversity of academic practices in publication processes. This study presents a comprehensive synthesis of the published literature documenting dated and undated archaeological materials from the Middle Stone Age and Later Stone Age in Northwest Africa, notably Morocco, Algeria, Tunisia and Libya. No previously published open-access database exists for these chronocultural periods in the region. Our dataset encompasses 993 sites and 1152 dates spanning approximately 370,000 to 8,000 years ago. Through a critical evaluation of the dates, we reveal qualitative and quantitative disparities and highlight the potential of the current archaeological record. While only ~10% of sites are dated and ~4.5% have reliable dates associated with a human occupation, this database holds significant potential for demographic and taxonomic meta-analyses as well as for methodological studies associated with chronological data in archaeology.(1) Overview Context Spatial Coverage Temporal Coverage (2) Methods Archaeological Site Geographical Coordinates Site Type And Fieldwork Methodology Cultural Label Chronology And Data Reliability Quality Control Constraints (3) Dataset Description (4) Reuse Potentia

FashionBrain Project: A Vision for Understanding Europe's Fashion Data Universe

A core business in the fashion industry is the understanding and prediction of customer needs and trends. Search engines and social networks are at the same time a fundamental bridge and a costly middleman between the customer’s purchase intention and the retailer. To better exploit Europe’s distinctive characteristics e.g., multiple languages, fashion and cultural differences, it is pivotal to reduce retailers’ dependence to search engines. This goal can be achieved by harnessing various data channels (manufacturers and distribution networks, online shops, large retailers, social media, market observers, call centers, press/magazines etc.) that retailers can leverage in order to gain more insight about potential buyers, and on the industry trends as a whole. This can enable the creation of novel on-line shopping experiences, the detection of influencers, and the prediction of upcoming fashion trends. In this paper, we provide an overview of the main research challenges and an analysis of the most promising technological solutions that we are investigating in the FashionBrain project