Spread-out percolation on transitive graphs of polynomial growth

Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in $G$. We show that the critical probability $p_c(G_r)$ for Bernoulli bond percolation on $G_r$ satisfies $p_c(G_r) \sim 1/\mathrm{deg}(G_r)$ as $r\to\infty$. This extends work of Penrose and Bollob\'as-Janson-Riordan, who considered the case $G=\mathbb{Z}^d$. Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces a new notion of dimension in groups. It also verifies a special case of a conjecture of Easo and Hutchcroft.Comment: 35 page

Schertz style class invariants for higher degree CM fields

Special values of Siegel modular functions for $\operatorname{Sp} (\mathbb{Z})$ generate class fields of CM fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie in the same class fields (class invariants) thus help to speed up the computation of those mathematical objects. We show that modular functions for the subgroup $\Gamma^0 (N)\subseteq \operatorname{Sp}(\mathbb{Z})$ yield class invariants under some splitting conditions on $N$, generalising results due to Schertz from classical modular functions to Siegel modular functions. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an \emph{$N$-system}. Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo $N$. We also examine conditions under which the minimal polynomial of a class invariant is real. Examples show that we may obtain class invariants that are much smaller than in previous constructions

Graphs for torus actions on oriented manifolds with isolated fixed points and classification in dimension 6

Let a torus act on a compact oriented manifold $M$ with isolated fixed points, with an additional mild assumption that its isotropy submanifolds are orientable. We associate a signed labeled multigraph encoding the fixed point data (weights and signs at fixed points and isotropy submanifolds) of the manifold. We study operations on $M$ and its multigraph, (self) connected sum and blow up, etc. When the circle group acts on a 6-dimensional $M$, we classify such a multigraph by proving that we can convert it into the empty graph by successively applying two types of operations. In particular, this classifies the fixed point data of any such manifold. We prove this by showing that for any such manifold, we can successively take equivariant connected sums at fixed points with itself, $\mathbb{CP}^3$, and 6-dimensional analogue $Z_1$ and $Z_2$ of the Hirzebruch surfaces (and these with opposite orientations) to a fixed point free action on a compact oriented 6-manifold. We also classify a multigraph for a torus action on a 4-dimensional $M$.Comment: Added the assumption on the orientability of isotropy submanifolds. This paper supercedes arXiv:2108.07560; main results are new, while including all results of the previous on

Noetherianity of twisted Zhu algebra and bimodules

In this paper we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu algebras and bimodules (and their $g$-twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study

Is Universal Broadband Service Impossible?

Broadband Internet service is widely expected to be the fundamental universal service for the 21st century. But more than a decade of national and international struggles to close the digital divide between broadband haves and have nots suggest that reaching global universality will be a very difficult task. This paper argues that the strong guarantees made by the current broadband paradigm - low latency and constant availability - are unnecessary obstacles to its adoption as an affordable and universal digital service. We show that there is nonetheless a plausible strategy for deploying a Basic Broadband service that does not require such guarantees and is able to offer, at reasonable cost, almost all the critical and valuable services and applications currently delivered over low latency broadband, synchronous telepresence excepted.Comment: Appeared in IEEE 19th International Conference on Mobile Ad Hoc and Smart Systems 202

Building Ocean Climate Emulators

The current explosion in machine learning for climate has led to skilled, computationally cheap emulators for the atmosphere. However, the research for ocean emulators remains nascent despite the large potential for accelerating coupled climate simulations and improving ocean forecasts on all timescales. There are several fundamental questions to address that can facilitate the creation of ocean emulators. Here we focus on two questions: 1) the role of the atmosphere in improving the extended skill of the emulator and 2) the representation of variables with distinct timescales (e.g., velocity and temperature) in the design of any emulator. In tackling these questions, we show stable prediction of surface fields for over 8 years, training and testing on data from a high-resolution coupled climate model, using results from four regions of the globe. Our work lays out a set of physically motivated guidelines for building ocean climate emulators

Graph-Skeleton: ~1% Nodes are Sufficient to Represent Billion-Scale Graph

Due to the ubiquity of graph data on the web, web graph mining has become a hot research spot. Nonetheless, the prevalence of large-scale web graphs in real applications poses significant challenges to storage, computational capacity and graph model design. Despite numerous studies to enhance the scalability of graph models, a noticeable gap remains between academic research and practical web graph mining applications. One major cause is that in most industrial scenarios, only a small part of nodes in a web graph are actually required to be analyzed, where we term these nodes as target nodes, while others as background nodes. In this paper, we argue that properly fetching and condensing the background nodes from massive web graph data might be a more economical shortcut to tackle the obstacles fundamentally. To this end, we make the first attempt to study the problem of massive background nodes compression for target nodes classification. Through extensive experiments, we reveal two critical roles played by the background nodes in target node classification: enhancing structural connectivity between target nodes, and feature correlation with target nodes. Followingthis, we propose a novel Graph-Skeleton1 model, which properly fetches the background nodes, and further condenses the semantic and topological information of background nodes within similar target-background local structures. Extensive experiments on various web graph datasets demonstrate the effectiveness and efficiency of the proposed method. In particular, for MAG240M dataset with 0.24 billion nodes, our generated skeleton graph achieves highly comparable performance while only containing 1.8% nodes of the original graph.Comment: 21 pages, 11 figures, In Proceedings of the ACM Web Conference 2024 (WWW'24

Pseudo-K\"ahler structure on the SL(3,R)\mathrm{SL}(3,\mathbb{R})-Hitchin component and Goldman symplectic form

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-K\"ahler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichm\"uller space. By comparing our symplectic form with Goldman's $\boldsymbol{\omega}_G$, we prove that the pair $(\boldsymbol{\omega}_G, \mathbf{I})$ cannot define a K\"ahler structure on the Hitchin component.Comment: Title and introduction changed. Added a result regarding Goldman symplectic for

Role of chemical potential at kinetic freeze-out using Tsallis non-extensive statistics in proton-proton collisions at the Large Hadron Collider

The charged-particle transverse momentum spectra ($p_{\rm T}$-spectra) measured by the ALICE collaboration for $pp$ collisions at $\sqrt {s} =$ 7 and 13 TeV have been studied using a thermodynamically consistent form of Tsallis non-extensive statistics. The Tsallis distribution function is fitted to the $p_{\rm T}$-spectra and the results are analyzed as a function of final state charged-particle multiplicity for various light flavor and strange particles, such as $\pi^{\pm}, K^{\pm}, p+\bar{p}, \phi, \Lambda+\bar{\Lambda}, \Xi+\bar{\Xi}, \Omega+\bar{\Omega}$. At the LHC energies, particles and antiparticles are produced in equal numbers. However, the equality of particle and antiparticle yields at the kinetic freeze-out may imply that they have the same but opposite chemical potential which is not necessarily zero. We use an alternative procedure that makes use of parameter redundancy, by introducing a finite chemical potential at the kinetic freeze-out stage. This article emphasizes the importance of the chemical potential of the system produced in $pp$ collisions at the LHC energies using the Tsallis distribution function which brings the system to a single freeze-out scenario.Comment: Same as the published version in EPJ

Misspecification-robust Sequential Neural Likelihood for Simulation-based Inference

Simulation-based inference techniques are indispensable for parameter estimation of mechanistic and simulable models with intractable likelihoods. While traditional statistical approaches like approximate Bayesian computation and Bayesian synthetic likelihood have been studied under well-specified and misspecified settings, they often suffer from inefficiencies due to wasted model simulations. Neural approaches, such as sequential neural likelihood (SNL) avoid this wastage by utilising all model simulations to train a neural surrogate for the likelihood function. However, the performance of SNL under model misspecification is unreliable and can result in overconfident posteriors centred around an inaccurate parameter estimate. In this paper, we propose a novel SNL method, which through the incorporation of additional adjustment parameters, is robust to model misspecification and capable of identifying features of the data that the model is not able to recover. We demonstrate the efficacy of our approach through several illustrative examples, where our method gives more accurate point estimates and uncertainty quantification than SNL