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We consider the community detection problem in sparse random hypergraphs. Angelini et al. (2015) conjectured the existence of a sharp threshold on model parameters for community detection in sparse hypergraphs generated by a hypergraph stochastic block model. We solve the positive part of the conjecture for the case of two blocks: above the threshold, there is a spectral algorithm which asymptotically almost surely constructs a partition of the hypergraph correlated with the true partition. Our method is a generalization to random hypergraphs of the method developed by Massouli\'{e} (2014) for sparse random graphs., Comment: 44 pages, 5 figures

We prove the sharp bound for the probability that two experts who have access to different information, represented by different $\sigma$-fields, will give radically different estimates of the probability of an event. This is relevant when one combines predictions from various experts in a common probability space to obtain an aggregated forecast. The optimizer for the bound is explicitly described. This paper was originally titled ‘Contradictory predictions’.

This article looks at the film No Smoking (2007), as an example of a ‘Kafkaesque’ piece of art which, though a commercial failure, attempts an ingenious critique of the absurdity of the neoliberal order (and impending fascism). The article examines how the film explores the promises (and their non-fulfilment) of urban space and technology, surveillance, censorship, the rise of right-wing populism, and the negotiations of identity in the irrational neoliberal order.

We consider random labelings of finite graphs conditioned on a small fixed number of peaks. We introduce a continuum framework where a combinatorial graph is associated with a metric graph and edges are identified with intervals. Next we consider a sequence of partitions of the edges of the metric graph with the partition size going to zero. As the mesh of the subdivision goes to zero, the conditioned random labelings converge, in a suitable sense, to a deterministic function which evolves as an increasing process of subsets of the metric graph that grows at rate one while maximizing an appropriate notion of entropy. We call such functions floodings. We present a number of qualitative and quantitative properties of floodings and some explicit examples.

Background The selection of appropriate kit and PCR equipment for the detection of SARS CoV-2 is critically important in view of many options available in the diagnostic market. Since last year many molecular products are available for COVID-19 diagnostics., some of these diagnostics have become commercially available for healthcare workers and clinical laboratories. However, the diagnostic technologies have specific limitations and reported several false-positive and false-negative cases, especially during the early stages of kit development and use. The current article addresses these and other relevant questions important to the medical microbiologists running or aspiring to run COVID diagnostic services using PCR and related technologies. Methods In this Systematic Review we follow Preferred Reporting Items for a Systematic Review and Meta-analysis of Diagnostic Test Accuracy Studies (PRISMA-DTA). A total of 258 citations retrieved, among those 77 peer reviewed articles was assessed for eligibility, and 181 studies were excluded. Based on inclusion criteria final data extraction was done. Results The question of diagnostic dilemma has also been addressed in view of discordant results between assays, inter-test variability, repeat testing requirements in specific settings and inconclusive or indeterminate results. Kit efficiency was satisfactory for all assays and the estimates varied within sample types and technology. Using clinical samples, we observed some variations in detection rate between kits. Importantly, none of the assays showed cross-reactivity with other respiratory (corona) viruses, except as expected for the SARS-CoV-1 E-gene. Conclusions We conclude SARS CoV-2 related molecular assays differed considerably in performance. Hence we need to understand importance of molecular diagnostics test interpretation in light of the latest pandemic virus.

Background The safety of training in off-pump coronary artery bypass (OPCAB) surgery and the stage at which trainees should be exposed to this technique remain controversial. This single-center retrospective study aimed to compare outcomes of OPCAB surgery in consultant and trainee cases. Methods Between 2014 and 2018, all isolated OPCAB operations performed under the care of a consultant surgeon (G.A.) were analyzed. Cases where a surgeon below consultant grade performed at least 70% of the distal anastomoses were designated as “trainee cases” with the remaining cases designated as “consultant cases.” The baseline characteristics of patients, perioperative data, and short-term outcomes were prospectively collated and analyzed. Results During the study period, 245 OPCAB cases were identified: 142 (58%) consultant and 103 (42%) trainee cases. The trainee cases were performed exclusively by trainees in the final 2 years of the UK National Cardiothoracic Training Program. Both trainee and consultant groups had low mortality with two perioperative deaths occurring in either group. The rates of serious postoperative complications including stroke (n = 1 vs. 2, p = 0.759), resternotomy for bleeding (n = 3 vs. 7, p = 0.431), and mediastinal infection (n = 2 vs. 3, p = 0.926) were low and not significantly different between the two groups. Patients operated on by trainees had a slightly longer hospital stay than those operated on by the consultant surgeon, although this did not reach statistical significance (9.9 vs. 7.9 days). Conclusions These results demonstrate comparable outcomes in OPCAB surgery between a consultant surgeon and trainees. This study supports the conclusion that training surgeons in OPCAB is appropriate for trainees in the final years of cardiac surgery training.

Consider a spectrally positive Stable$(1\!+\!\alpha )$ process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning “sizes” varying during the lifetime. As for Crump–Mode–Jagers processes (with “characteristics”), we consider for each level the collection of individuals alive. We arrange their “sizes” at the crossing height from left to right to form an interval partition. We study the continuity and Markov properties of the interval-partition-valued process indexed by level. From the perspective of the Stable$(1\!+\!\alpha )$ process, this yields new theorems of Ray–Knight-type. From the perspective of branching processes, this yields new, self-similar models with dense sets of birth and death times of (mostly short-lived) individuals. This paper feeds into projects resolving conjectures by Feng and Sun (2010) on the existence of certain measure-valued diffusions with Poisson–Dirichlet stationary laws, and by Aldous (1999) on the existence of a continuum-tree-valued diffusion.

We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters $(\alpha,0)$ and $(\alpha,\alpha)$. The construction has two steps. The first is a general construction of interval partition processes obtained previously, by decorating the jumps of a L\'evy process with independent excursions. Here, we focus on the second step, which requires explicit transition kernels and what we call pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper, we establish connections to diffusions on decreasing sequences introduced by Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step towards resolving longstanding conjectures by Feng and Sun on measure-valued Poisson-Dirichlet diffusions, and by Aldous on a continuum-tree-valued diffusion., Comment: 50 pages, 8 figures. Following arXiv:1909.02584 [math.PR], this is the second in a sequence of three papers that will collectively supersede arXiv:1609.06706 [math.PR]

We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with alpha-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths., 12 pages; this preprint generalises and supersedes the parts of arXiv:1609.06706 concerning topologies on interval partitions

We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusions with suitable drifts and diffusions. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results., 10 pages. Keywords: Reflected Brownian motion, Wasserstein distance, relative entropy, transportation cost-information inequality, concentration of measure, competing Brownian particles

We establish two results about local times of spectrally positive stable processes. The first is a general approximation result, uniform in space and on compact time intervals, in a model where each jump of the stable process may be marked by a random path. The second gives moment control on the H\"older constant of the local times, uniformly across a compact spatial interval and in certain random time intervals. For the latter, we introduce the notion of a L\'evy process restricted to a compact interval, which is a variation of Lambert's L\'evy process confined in a finite interval and of Pistorius' doubly reflected process. We use the results of this paper to exhibit a class of path-continuous branching processes of Crump-Mode-Jagers type with continuum genealogical structure. A further motivation for this study lies in the construction of diffusion processes in spaces of interval partitions and R-trees, which we explore in forthcoming articles., Comment: 27 pages

Consider the Aldous Markov chain on the space of rooted binary trees with $n$ labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix $1\le k < n$ and project the leaf mass onto the subtree spanned by the first $k$ leaves. This yields a binary tree with edge weights that we call a "decorated $k$-tree with total mass $n$." We introduce label swapping dynamics for the Aldous chain so that, when it runs in stationarity, the decorated $k$-trees evolve as Markov chains themselves, and are projectively consistent over $k\le n$. The construction of projectively consistent chains is a crucial step in the construction of the Aldous diffusion on continuum trees by the present authors, which is the $n\rightarrow \infty$ continuum analogue of the Aldous chain and will be taken up elsewhere. Some of our results have been generalized to Ford's alpha model trees., 30 pages, 8 figures

We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost function is the square of the Euclidean distance and corresponds to the cumulant generating function of the multivariate standard normal distribution. The optimal coupling is usually described via an extended notion of convex/concave functions and their gradient maps. These extended notions are nonintuitive and do not satisfy useful inequalities such as Jensen's inequality. Under mild regularity conditions, we show that all such extended gradient maps can be recovered as the usual supergradients of a nonnegative concave function on the space of probability distributions. This embedding provides a universal geometry for all such optimal transports and an unexpected connection with information geometry of exponential families of distributions., Comment: 8 pages, to appear in Indian Journal of Pure and Applied Mathematics (Special volume dedicated to Prof. B. V. Rao)

Gold nanoparticles have unique properties that are highly dependent on their shape and size. Synthetic methods that enable precise control over nanoparticle morphology currently require shape-directing agents such as surfactants or polymers that force growth in a particular direction by adsorbing to specific crystal facets. These auxiliary reagents passivate the nanoparticles' surface, and thus decrease their performance in applications like catalysis and surface-enhanced Raman scattering. Here, a surfactant- and polymer-free approach to achieving high-performance gold nanoparticles is reported. A theoretical framework to elucidate the growth mechanism of nanoparticles in surfactant-free media is developed and it is applied to identify strategies for shape-controlled syntheses. Using the results of the analyses, a simple, green-chemistry synthesis of the four most commonly used morphologies: nanostars, nanospheres, nanorods, and nanoplates is designed. The nanoparticles synthesized by this method outperform analogous particles with surfactant and polymer coatings in both catalysis and surface-enhanced Raman scattering.

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n., Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Fields

This note proves a weak sharpness principle as conjectured by Gneiting et al. (2007) in connection with probabilistic forecasting subject to calibration constraints. Copyright 2009, Oxford University Press.

We consider the following problem in stochastic portfolio theory. Are there portfolios that are relative arbitrages with respect to the market portfolio over very short periods of time under realistic assumptions? We answer a slightly relaxed question affirmative in the following high dimensional sense, where dimension refers to the number of stocks being traded. Very roughly, suppose that for every dimension we have a continuous semimartingale market such that (i) the vector of market weights in decreasing order has a stationary regularly varying tail with an index between $-1$ and $-1/2$ and (ii) zero is not a limit point of the relative volatilities of the stocks. Then, given a probability $\eta < 1$ arbitrarily close to one, two arbitrarily small $\epsilon, \delta >0$, and an arbitrarily high positive amount $M$, for all high enough dimensions, it is possible to construct a functionally generated portfolio such that, with probability at least $\eta$, its relative value with respect to the market at time $\delta$ is at least $M$, and never goes below $(1-\epsilon)$ during $[0, \delta]$. There are two phase transitions; if the index of the tail is less than $-1$ or larger than $-1/2$. The construction uses properties of regular variation, high-dimensional convex geometry and concentration of measure under Dirichlet distributions. We crucially use the notion of $(K,N)$ convex functions introduced by Erbar, Kuwada, Sturm in the context of curvature-dimension conditions and Bochner's inequalities., Comment: 31 pages, 5 figures, minor editing

We consider a trader who wants to direct his or her portfolio towards a set of acceptable wealths given by a convex risk measure. We propose a Monte Carlo algorithm, whose inputs are the joint law of stock prices and the convex risk measure, and whose outputs are the numerical values of initial capital requirement and the functional form of a trading strategy for achieving acceptability. We also prove optimality of the capital obtained. Explicit theoretical evaluations of hedging strategies are extremely difficult, and we avoid the problem by resorting to such computational methods. The main idea is to utilize the finite Vapnik–C˘ervonenkis dimension of a class of possible strategies.

We study a model of mass redistribution on a finite graph. We address the questions of convergence to equilibrium and the rate of convergence. We present theorems on the distribution of empty sites and the distribution of mass at a fixed vertex. These distributions are related to random permutations with certain peak sets., Comment: Published at http://dx.doi.org/10.1214/14-AAP1035 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using transportation cost inequalities for stochastic processes we provide uniform fluctuation bounds for the ordered particles, their local time of collisions and various associated statistics over intervals of time. For example, such processes, when exponentiated and rescaled, exhibit power law decay under stationarity; we derive concentration bounds for the empirical estimates of the index of the power law over large intervals of time. A key ingredient in our proofs is a novel upper bound on the Lipschitz constant of the Skorokhod map that transforms a multidimensional Brownian path to a path which is constrained not to leave the positive orthant.

Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic (in $n$) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These processes track the evolution of short cycles in the graph. If we now take $d$ to infinity, certain GFF-like properties emerge., Comment: Published in at http://dx.doi.org/10.1214/13-AOP864 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Consider an equity market with $n$ stocks. The vector of proportions of the total market capitalizations that belong to each stock is called the market weight. The market weight defines the market portfolio which is a buy-and-hold portfolio representing the performance of the entire stock market. Consider a function that assigns a portfolio vector to each possible value of the market weight, and we perform self-financing trading using this portfolio function. We study the problem of characterizing functions such that the resulting portfolio will outperform the market portfolio in the long run under the conditions of diversity and sufficient volatility. No other assumption on the future behavior of stock prices is made. We prove that the only solutions are functionally generated portfolios in the sense of Fernholz. A second characterization is given as the optimal maps of a remarkable optimal transport problem. Both characterizations follow from a novel property of portfolios called multiplicative cyclical monotonicity., 31 pages, 5 figures; substantially revised; Section 4 illustrates the optiaml transport approach with empirical examples

A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider $d$ many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree $2d$. We consider the following dynamics. The dimension (i.e. size) of each permutation grows by coupled Chinese Restaurant Processes, while in time each permutation evolves according to the random transposition chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. In dimension, it was shown by Johnson and Pal (2014) that cycle counts are described by a Poisson field of Yule processes. Here, we give a Poisson random surface description in dimension and time of the limiting cycle counts for every $d$. As $d$ grows to infinity, the fluctuation of the limiting cycle counts, across dimension, converges to the Gaussian Free Field. In time this field is preserved by a stationary Gaussian dynamics. The laws of these processes are similar to eigenvalue fluctuations of the minor process of a real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationary stochastic processes., Comment: 43 pages, 6 figures. arXiv admin note: text overlap with arXiv:1406.7043 by other authors

We prove that the distance between two reflected Brownian motions, driven by the same white noise, outside a sphere in a 3-dimensional flat torus does not converge to 0, a.s., if the radius of the sphere is sufficiently small, relative to the size of the torus., Published in at http://dx.doi.org/10.1214/12-AOP831 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study a family of n-dimensional diffusions, taking values in the unit simplex of vectors with nonnegative coordinates that add up to one. These processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher diffusions, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions, which have been recently introduced by Going-Jaeschke and Yor., Comment: Published in at http://dx.doi.org/10.1214/11-AOP704 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized., Comment: Published in at http://dx.doi.org/10.1214/11-AOP673 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a Fleming-Viot-type particle system consisting of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches. If there are only two particles and the underlying motion is a Bessel process on $(0,\infty)$, both particles converge to 0 at a finite time if and only if the dimension of the Bessel process is less than 0. If the underlying diffusion is Brownian motion with a drift stronger than (but arbitrarily close to, in a suitable sense) the drift of a Bessel process, all particles converge to 0 at a finite time, for any number of particles.

We consider a family of hard core objects moving as independent Brownian motions confined to a vessel by reflection. These are subject to gravitational forces modeled by drifts. The stationary distribution for the process has many interesting implications, including an illustration of the Archimedes' principle. The analysis rests on constructing reflecting Brownian motion with drift in a general open connected domain and studying its stationary distribution. In dimension two we utilize known results about sphere packing., Published in at http://dx.doi.org/10.1214/11-AAP765 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)