Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

Spectrum of non-Hermitian heavy tailed random matrices

Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu_alpha. In contrast with the Hermitian case, we find that mu_alpha is not heavy tailed.Comment: Expanded version of a paper published in Communications in Mathematical Physics 307, 513-560 (2011

Spectral density of random graphs with topological constraints

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting spectral density.Comment: 24 pages, 6 figure

A real quaternion spherical ensemble of random matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB, where \bA and \bB are independent $N\times N$ matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices $\beta=1,2,4$. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure \bY= \bA^{-1} \bB, where \bA and \bB are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.Comment: 25 pages, 3 figures. Added another citation in version

Spectral Theory of Sparse Non-Hermitian Random Matrices

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.Comment: 60 pages, 10 figure

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

Thermal modelling of gas generation and retention in the Jurassic organic-rich intervals in the Darquain field, Abadan Plain, SW Iran

The petroleum system with Jurassic source rocks is an important part of the hydrocarbons discovered in the Middle East. Limited studies have been done on the Jurassic intervals in the 26,500 km2 Abadan Plain in south-west Iran, mainly due to the deep burial and a limited number of wells that reach the basal Jurassic successions. The goal of this study was to evaluate the Jurassic organic-rich intervals and shale gas play in the Darquain field using organic geochemistry, organic petrography, biomarker analysis, and basin modelling methods. This study showed that organic-rich zones present in the Jurassic intervals of Darquain field could be sources of conventional and unconventional gas reserves. The organic matter content of samples from the organic-rich zones corresponds to medium-to-high-sulphur kerogen Type II-S marine origin. The biomarker characteristics of organic-rich zones indicate carbonate source rocks that contain marine organic matter. The biomarker results also suggest a marine environment with reducing conditions for the source rocks. The constructed thermal model for four pseudo-wells indicates that, in the kitchen area of the Jurassic gas reserve, methane has been generated in the Sargelu and Neyriz source rocks from Early Cretaceous to recent times and the transformation ratio of organic matter is more than 97%. These organic-rich zones with high initial total organic carbon (TOC) are in the gas maturity stage [1.5–2.2% vitrinite reflectance in oil (Ro)] and could be good unconventional gas reserves and gas source rocks. The model also indicates that there is a huge quantity of retained gas within the Jurassic organic-rich intervals

The effect of hypoxia and work intensity on insulin resistance in type 2 diabetes

Context:Hypoxia and muscle contraction stimulate glucose transport in vitro. We have previously demonstrated that exercise and hypoxia have an additive effect on insulin sensitivity in type 2 diabetics.Objectives:Our objective was to examine the effects of three different hypoxic/exercise (Hy Ex) trials on glucose metabolism and insulin resistance in the 48 h after acute hypoxia in type 2 diabetics.Design, Participants, and Interventions:Eight male type 2 diabetics completed 60 min of hypoxic [mean (sem) O(2) = ∼14.7 (0.2)%] exercise at 90% of lactate threshold [Hy Ex(60); 49 (1) W]. Patients completed an additional two hypoxic trials of equal work, lasting 40 min [Hy Ex(40); 70 (1) W] and 20 min [Hy Ex(20); 140 (12) W].Main Outcome Measures:Glucose rate of appearance and rate of disappearance were determined using the one-compartment minimal model. Homeostasis models of insulin resistance (HOMA(IR)), fasting insulin resistance index and β-cell function (HOMA(β-cell)) were calculated at 24 and 48 h after trials.Results:Peak glucose rate of appearance was highest during Hy Ex(20) [8.89 (0.56) mg/kg · min, P < 0.05]. HOMA(IR) and fasting insulin resistance index were improved in the 24 and 48 h after Hy Ex(60) and Hy Ex(40) (P < 0.05). HOMA(IR) decreased 24 h after Hy Ex(20) (P < 0.05) and returned to baseline values at 48 h.Conclusions:Moderate-intensity exercise in hypoxia (Hy Ex(60) and Hy Ex(40)) stimulates acute- and moderate-term improvements in insulin sensitivity that were less apparent in Hy Ex(20). Results suggest that exercise duration and not total work completed has a greater influence on acute and moderate-term glucose control in type 2 diabetics