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Free, publicly-accessible full text available June 1, 2025
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We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the r-uniform Erdo ̋s–Rényi hypergraph for any fixed r≥2, generalizing and improving on previous results for the Erdo ̋s–Rényi graph (r=2). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts.more » « lessFree, publicly-accessible full text available April 1, 2025
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Proving the “expectation-threshold” conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property
on a finite set , \[\] whereand are the threshold and “expectation threshold” of , and is the maximum of and the maximum size of a minimal member of . -
We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $\mu$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities, which guarantee entropy contraction not for all distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: $\mu$ is entropically independent exactly when a transformed version of the generating polynomial of $\mu$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain (1) tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions, (2) the tight mixing time of $O(n\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$, and (3) nearly-linear time $\widetilde O_{\delta}(n)$ samplers for the hardcore and Ising models on $n$-node graphs that have $\delta$-relative gap to the tree-uniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree $\Delta$ of the graph, and is therefore optimal even for high-degree graphs, and in fact, is sublinear in the size of the graph for high-degree graphs.more » « less
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Abstract A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $\mathbb{F}_q^{\,n}$ is asymptotically (as $n \to \infty$) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of $\mathbb{F}_q^{\,n}$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.more » « less
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Abstract Assistive wearable soft robotic systems have recently made a surge in the field of biomedical robotics, as soft materials allow safe and transparent interactions between the users and devices. A recent interest in the field of soft pneumatic actuators (SPAs) has been the introduction of a new class of actuators called fabric soft pneumatic actuators (FSPAs). These actuators exploit the unique capabilities of different woven and knit textiles, including zero initial stiffness, full collapsibility, high power-to-weight ratio, puncture resistant, and high stretchability. By using 2D manufacturing methods we are able to create actuators that can extend, contract, twist, bend, and perform a combination of these motions in 3D space. This paper presents a comprehensive simulation and design tool for various types of FSPAs using finite element method (FEM) models. The FEM models are developed and experimentally validated, in order to capture the complex non-linear behavior of individual actuators optimized for free displacement and blocked force, applicable for wearable assistive tasks.