Local Kesten–McKay Law for Random Regular Graphs

We study the adjacency matrices of random $d$-regular graphs with large but fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$ down to the optimal spectral scale, we prove that the Green's functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten--McKay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. Our method is based on estimating the Green's function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs

Edge rigidity and universality of random regular graphs of intermediate degree

For random $d$-regular graphs on $N$ vertices with $1 \ll d \ll N^{2/3}$, we develop a $d^{-1/2}$ expansion of the local eigenvalue distribution about the Kesten-McKay law up to order $d^{-3}$. This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erd\H{o}s-R\'enyi graphs of the same average degree. As a first application, for $1 \ll d \ll N^{2/3}$, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by $(2 + o(1)) \sqrt{d - 1}$. As a second application, for $N^{2/9} \ll d \ll N^{1/3}$, we prove that the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have second-largest eigenvalue strictly less than $2 \sqrt{d - 1}$.The work of J.H. is supported by the Institute for Advanced Study. A.K. gratefully acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715539_RandMat) and from the Swiss National Science Foundation through the SwissMAP grant. The work of H.-T.Y. is partially supported by NSF Grants DMS-1606305 and DMS-1855509, and a Simons Investigator award

Edge rigidity and universality of random regular graphs of intermediate degree

© 2020, Springer Nature Switzerland AG. For random d-regular graphs on N vertices with 1 ≪ d≪ N2 / 3, we develop a d- 1 / 2 expansion of the local eigenvalue distribution about the Kesten–McKay law up to order d- 3. This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erdős–Rényi graphs of the same average degree. As a first application, for 1 ≪ d≪ N2 / 3, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by (2+o(1))d-1. As a second application, for N2 / 9≪ d≪ N1 / 3, we prove that the extremal eigenvalues are concentrated at scale N- 2 / 3 and their fluctuations are governed by Tracy–Widom statistics. Thus, in the same regime of d, 52 % of all d-regular graphs have second-largest eigenvalue strictly less than 2d-1.The work of J.H. is supported by the Institute for Advanced Study. A.K. gratefully acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715539_RandMat) and from the Swiss National Science Foundation through the SwissMAP grant. The work of H.-T.Y. is partially supported by NSF Grants DMS-1606305 and DMS-1855509, and a Simons Investigator award

Bulk eigenvalue statistics for random regular graphs

We consider the uniform random $d$-regular graph on $N$ vertices, with $d \in [N^\alpha, N^{2/3-\alpha}]$ for arbitrary $\alpha > 0$. We prove that in the bulk of the spectrum the local eigenvalue correlation functions and the distribution of the gaps between consecutive eigenvalues coincide with those of the Gaussian Orthogonal Ensemble.AK was partly supported by Swiss National Science Foundation grant 144662. HTY was partly supported by NSF grant DMS-1307444 and a Simons Investigator fellowship.his is the author accepted manuscript. It is currently under an indefinite embargo pending publication by the Institute of Mathematical Statistics

Cathelicidin preserves intestinal barrier function in polymicrobial sepsis.

ObjectivesThe intestinal epithelium compartmentalizes the sterile bloodstream and the commensal bacteria in the gut. Accumulating evidence suggests that this barrier is impaired in sepsis, aggravating systemic inflammation. Previous studies reported that cathelicidin is differentially expressed in various tissues in sepsis. However, its role in sepsis-induced intestinal barrier dysfunction has not been investigated.DesignTo examine the role of cathelicidin in polymicrobial sepsis, cathelicidin wild-(Cnlp+/+) and knockout (Cnlp-/-) mice underwent cecal-ligation and puncture (CLP) followed by the assessment of septic mortality and morbidity as well as histological, biochemical, immunological, and transcriptomic analyses in the ileal tissues. We also evaluated the prophylactic and therapeutic efficacies of vitamin D3 (an inducer of endogenous cathelicidin) in the CLP-induced murine polymicrobial sepsis model.ResultsThe ileal expression of cathelicidin was increased by three-fold after CLP, peaking at 4 h. Knockout of Cnlp significantly increased 7-day mortality and was associated with a higher murine sepsis score. Alcian-blue staining revealed a reduced number of mucin-positive goblet cells, accompanied by reduced mucin expression. Increased number of apoptotic cells and cleavage of caspase-3 were observed. Cnlp deletion increased intestinal permeability to 4kD fluorescein-labeled dextran and reduced the expression of tight junction proteins claudin-1 and occludin. Notably, circulating bacterial DNA load increased more than two-fold. Transcriptome analysis revealed upregulation of cytokine/inflammatory pathway. Depletion of Cnlp induced more M1 macrophages and neutrophils compared with the wild-type mice after CLP. Mice pre-treated with cholecalciferol (an inactive form of vitamin D3) or treated with 1alpha, 25-dihydroxyvitamin D3 (an active form of VD3) had decreased 7-day mortality and significantly less severe symptoms. Intriguingly, the administration of cholecalciferol after CLP led to worsened 7-day mortality and the associated symptoms.ConclusionsEndogenous cathelicidin promotes intestinal barrier integrity accompanied by modulating the infiltration of neutrophils and macrophages in polymicrobial sepsis. Our data suggested that 1alpha, 25-dihydroxyvitamin D3 but not cholecalciferol is a potential therapeutic agent for treating sepsis

Occupation times of long-range exclusion and connections to KPZ class exponents

With respect to a class of long-range exclusion processes on \ZZ^d, with single particle transition rates of order $|\cdot|^{-(d+\alpha)}$, starting under Bernoulli invariant measure $\nu_\rho$ with density $\rho$, we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on $\alpha$, $d$ and $\rho$ with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter $H\in [1/2,3/4]$. However, in the asymmetric case, we study the asymptotics of the variances, which when $d=1$ and $\rho=1/2$ points to a curious dichotomy between long-range strength parameters $03/2$. In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short-range model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.The research of CB was supported in part by the French Ministry of Education through the grant ANR JCJC EDNHS. PG thanks FCT (Portugal) for support through the research project PTDC/MAT/109844/2009 and CNPq (Brazil) for support through the research project 480431/2013-2. PG thanks CMAT for support by "FEDER" through the "Programa Operacional Factores de Competitividade COMPETE" and by FCT through the project PEst-C/MAT/UI0013/2011. SS was supported in part by ARO grant W911NF-14-1-0179

Pan-cancer analysis of whole genomes

Cancer is driven by genetic change, and the advent of massively parallel sequencing has enabled systematic documentation of this variation at the whole-genome scale(1-3). Here we report the integrative analysis of 2,658 whole-cancer genomes and their matching normal tissues across 38 tumour types from the Pan-Cancer Analysis of Whole Genomes (PCAWG) Consortium of the International Cancer Genome Consortium (ICGC) and The Cancer Genome Atlas (TCGA). We describe the generation of the PCAWG resource, facilitated by international data sharing using compute clouds. On average, cancer genomes contained 4-5 driver mutations when combining coding and non-coding genomic elements; however, in around 5% of cases no drivers were identified, suggesting that cancer driver discovery is not yet complete. Chromothripsis, in which many clustered structural variants arise in a single catastrophic event, is frequently an early event in tumour evolution; in acral melanoma, for example, these events precede most somatic point mutations and affect several cancer-associated genes simultaneously. Cancers with abnormal telomere maintenance often originate from tissues with low replicative activity and show several mechanisms of preventing telomere attrition to critical levels. Common and rare germline variants affect patterns of somatic mutation, including point mutations, structural variants and somatic retrotransposition. A collection of papers from the PCAWG Consortium describes non-coding mutations that drive cancer beyond those in the TERT promoter(4); identifies new signatures of mutational processes that cause base substitutions, small insertions and deletions and structural variation(5,6); analyses timings and patterns of tumour evolution(7); describes the diverse transcriptional consequences of somatic mutation on splicing, expression levels, fusion genes and promoter activity(8,9); and evaluates a range of more-specialized features of cancer genomes(8,10-18).Peer reviewe

Search for pair production of Higgs bosons in the bb¯bb¯ final state using proton-proton collisions at s√=13 TeV with the ATLAS detector

A search for Higgs boson pair production in the bb¯¯bb¯¯ final state is carried out with up to 36.1 fb−1 of LHC proton-proton collision data collected at s√=13 TeV with the ATLAS detector in 2015 and 2016. Three benchmark signals are studied: a spin-2 graviton decaying into a Higgs boson pair, a scalar resonance decaying into a Higgs boson pair, and Standard Model non-resonant Higgs boson pair production. Two analyses are carried out, each implementing a particular technique for the event reconstruction that targets Higgs bosons reconstructed as pairs of jets or single boosted jets. The resonance mass range covered is 260–3000 GeV. The analyses are statistically combined and upper limits on the production cross section of Higgs boson pairs times branching ratio to bb¯¯bb¯¯ are set in each model. No significant excess is observed; the largest deviation of data over prediction is found at a mass of 280 GeV, corresponding to 2.3 standard deviations globally. The observed 95% confidence level upper limit on the non-resonant production is 13 times the Standard Model prediction