Logarithmic corrections to the Alexander-Orbach conjecture for the four-dimensional uniform spanning tree

We compute the precise logarithmic corrections to Alexander-Orbach behaviour for various quantities describing the geometric and spectral properties of the four-dimensional uniform spanning tree. In particular, we prove that the volume of an intrinsic $n$-ball in the tree is $n^2 (\log n)^{-1/3+o(1)}$, that the typical intrinsic displacement of an $n$-step random walk is $n^{1/3} (\log n)^{1/9-o(1)}$, and that the $n$-step return probability of the walk decays as $n^{-2/3}(\log n)^{1/9-o(1)}$.Comment: 41 page

Scaling limits and universality: Critical percolation on weighted graphs converging to an L3L^3 graphon

We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdos-Renyi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$ graphon. In particular, we define a notion of the critical window in this setting. The $L^3$ assumption ensures that the model is in the Erdos-Renyi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel's RGIV model [9] inside the critical window. Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs [52]. In [10], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Rucinski and Wormald's random graph process with degree restrictions [56] provided an additional technical condition about the barely subcritical behavior of this model can be proved.Comment: 65 pages, 1 figure, the universality principle (Theorem 3.4) from arXiv:1411.3417 has now been included in this paper. v2: minor change

Investigations and Analysis of Dynamical and Steady State Properties of Chemical Reaction Systems

In this paper, we investigate results from chemical reaction network theory and a list of techniques to test for the reaction-coordinates dynamical system to have a partial order induced by a positive orthant cone. A successful result from one of these tests guarantees mono-stationarity (and indeed convergence). We also investigate a recently published algorithmic and computational approach to determine whether a reaction network establishes mono- or multi-stationarity. We test new reactions that have not been previously introduced in the literature for mono- or multi-stationarity using this approach. This includes the two-site phosphorylation reaction network and a modified double phosphorylation reaction network that more accurately models the action of the enzymes of two distinct sites. We also use the enzymatic futile cycle as a running example to illustrate these results. We conclude the two-site phosphorylation reaction network is multi-stationary; while the original double phosphorylation reaction network is also multi-stationary, our modified version is mono-stationary

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

We give a new characterization of the vertical-strip LLT polynomials $\mathrm{LLT}_P(x;q)$ as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if $q$ is replaced by $q+1$, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.Comment: 49 pages. This version has updated .bib, and some improvements in section

Towards Real-Time Simulation Of Hyperelastic Materials

We propose a new method for physics-based simulation supporting many different types of hyperelastic materials from mass-spring systems to three-dimensional finite element models, pushing the performance of the simulation towards real-time. Fast simulation methods such as Position Based Dynamics exist, but support only limited selection of materials; even classical materials such as corotated linear elasticity and Neo-Hookean elasticity are not supported. Simulation of these types of materials currently relies on Newton\u27s method, which is slow, even with only one iteration per timestep. In this work, we start from simple material models such as mass-spring systems or as-rigid-as-possible materials. We express the widely used implicit Euler time integration as an energy minimization problem and introduce auxiliary projection variables as extra unknowns. After our reformulation, the minimization problem becomes linear in the node positions, while all the non-linear terms are isolated in individual elements. We then extend this idea to efficiently simulate a more general spatial discretization using finite element method. We show that our reformulation can be interpreted as a quasi-Newton method. This insight enables very efficient simulation of a large class of hyperelastic materials. The quasi-Newton interpretation also allows us to leverage ideas from numerical optimization. In particular, we show that our solver can be further accelerated using L-BFGS updates (Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm). Our final method is typically more than ten times faster than one iteration of Newton\u27s method without compromising quality. In fact, our result is often more accurate than the result obtained with one iteration of Newton\u27s method. Our method is also easier to implement, implying reduced software development costs

Complex and Adaptive Dynamical Systems: A Primer

An thorough introduction is given at an introductory level to the field of quantitative complex system science, with special emphasis on emergence in dynamical systems based on network topologies. Subjects treated include graph theory and small-world networks, a generic introduction to the concepts of dynamical system theory, random Boolean networks, cellular automata and self-organized criticality, the statistical modeling of Darwinian evolution, synchronization phenomena and an introduction to the theory of cognitive systems. It inludes chapter on Graph Theory and Small-World Networks, Chaos, Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean Networks, Cellular Automata and Self-Organized Criticality, Darwinian evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements of Cognitive System Theory.Comment: unformatted version of the textbook; published in Springer, Complexity Series (2008, second edition 2010

Random matrices

138 pages, based on lectures by Bertrand Eynard at IPhT, SaclayWe provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals