A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models

Higher-order fluctuations in dense random graph models

Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. The main motivation to investigate these statistics is the fact that they are key to understanding fluctuations of regular subgraph counts -- the cornerstone of dense graph limit theory -- since they act as an orthogonal basis of a corresponding $L_2$ space. We also identify the resulting limiting Gaussian stochastic measures by means of the theory of generalised U-statistics and Gaussian Hilbert spaces, which we think is a suitable framework to describe and understand higher-order fluctuations in dense random graph models. With this article, we believe we answer the question "What is the central limit theorem of dense graph limit theory?".Comment: 28 page

On stochastic properties between some ordered random variables

A great number of articles have dealt with stochastic comparisons of ordered random variables in the last decades. In particular, distributional and stochastic properties of ordinary order statistics have been studied extensively in the literature. Sequential order statistics are proposed as an extension of ordinary order statistics. Since sequential order statistics models unify various models of ordered random variables, it is interesting to study their distributional and stochastic properties. In this work, we consider the problem of comparing sequential order statistics according to magnitude and location orders.Stochastic orderings, Reliability, Order statistics

Random pinning in glassy spin models with plaquette interactions

We use a random pinning procedure to study amorphous order in two glassy spin models. On increasing the concentration of pinned spins at constant temperature, we find a sharp crossover (but no thermodynamic phase transition) from bulk relaxation to localisation in a single state. At low temperatures, both models exhibit scaling behaviour. We discuss the growing length and time scales associated with amorphous order, and the fraction of pinned spins required to localize the system in a single state. These results, obtained for finite dimensional interacting models, provide a theoretical scenario for the effect of random pinning that differs qualitatively from previous approaches based either on mean-field, mode-coupling, or renormalization group reatments.Comment: 15 pages, 9 fig

A unified picture of ferromagnetism, quasi-long range order and criticality in random field models

By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long range order (QLRO) and criticality in the $d$-dimensional random field O(N) model in the whole ($N$, $d$) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with however no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely a topologically ordered "Bragg glass" phase, is absent in the 3--dimensional random field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around $d=4$.Comment: 4 pages, 4 figure

Encoding Sequential Information in Vector Space Models of Semantics: Comparing Holographic Reduced Representation and Random Permutation

Encoding information about the order in which words typically appear has been shown to improve the performance of high-dimensional semantic space models. This requires an encoding operation capable of binding together vectors in an order-sensitive way, and efficient enough to scale to large text corpora. Although both circular convolution and random permutations have been enlisted for this purpose in semantic models, these operations have never been systematically compared. In Experiment 1 we compare their storage capacity and probability of correct retrieval; in Experiments 2 and 3 we compare their performance on semantic tasks when integrated into existing models. We conclude that random permutations are a scalable alternative to circular convolution with several desirable properties