We establish necessary and sufficient conditions for consistent root reconstruction in continuous-time Markov models with countable state space on bounded-height trees. Here a root state estimator is said to be consistent if the probability that it returns to the true root state converges to 1 as the number of leaves tends to infinity. We also derive quantitative bounds on the error of reconstruction. Our results answer a question of Gascuel and Steel [GS10] and have implications for ancestral sequence reconstruction in a classical evolutionary model of nucleotide insertion and deletion [TKF91].

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter β, which is the natural dynamic associated to the usual quantum sine- Gordon model. It is shown that when β2 ∈ (0, 16π ) the Wick renormalised equation 3 is well-posed. In the regime β2 ∈ (0,4π), the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for β2 ∈ [4π, 16π ), the solution theory is provided via the theory of regularity structures 3 [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of 2 + 1-dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al.

We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.

We consider a general class of $N\times N$ random matrices whose entries $h_{ij}$ are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [14] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, $\max_{i,j} \E \abs{h_{ij}}^2$. As a consequence, we prove the universality of the local $n$-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width $W\gg N^{1-\e_n}$ with some $\e_n>0$ and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17,19,6].

In a uniform random recursive$k$-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to$n$, chooses$k$uniform random parents from among the nodes of smaller index. If$S$_{$n$}is the shortest path distance from node$n$to the root, then we determine the constant$σ$such that$S$_{$n$}/log$n$→$σ$in probability as$n$→∞. We also show that max_{1≤$i$≤$n$}$S$_{$i$}/log$n$→$σ$in probability.

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [Theor. Pop. Biol. 73 (2008) 158–170]. The first step, which is easily accomplished using results of Mueller and Tribe [Probab. Theory Related Fields 102 (1995) 519–545], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright–Fisher SPDE. A simple extension of a result of Durrett and Restrepo [Ann. Appl. Probab. 18 (2008) 334–358] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [Phys. Rev. E (3) 76 (2007) 041104, 20] have conjectured that genealogies in models of this type are described by the Bolthausen–Sznitman coalescent, see [Proc. Natl. Acad. Sci. USA 110 (2013) 437–442]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns “tracer dynamics” in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [In Stochastic Processes in Physics and Engineering (Bielefeld, 1986) (1988) 345–355 Reidel] developed for the Wright–Fisher SPDE is the key to the proof of that result.

We consider the KPZ equation in one space dimension driven by a sta- tionary centred space–time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf–Cole solution to the KPZ equation driven by space–time Gaussian white noise. While the limit- ing process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.

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We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions (⊖⊕), (⊖ free) and (free free), and the alternating interior arm exponents.

We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf–Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf–Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.

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We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface I. This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.

We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless, if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M times N$ matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $langle v, (X^* X - z)^{-1} w rangle - langle v,w rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v, w in mathbb C^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $Im z geq N^{-1+epsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$ with ||z|−1|>c for any c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $|z|-1=\oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $|z|-1=\oo(1)$ up to scale $N^{-1/4+ \e}$.

Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $\nu_{ij}$ with zero expectation and with variance $\sigma_{ij}^2$. We assume that the variances satisfy the normalization condition $\sum_{i} \sigma^2_{ij} = 1$ for all $j$ and that there is a positive constant $c$ such that $c\le N \sigma_{ij}^2 \le c^{-1}$. We further assume that the probability distributions $\nu_{ij}$ have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order $ (N \eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If $\gamma_j =\gamma_{j,N}$ denotes the {\it classical location} of the $j$-th eigenvalue under the semicircle law ordered in increasing order, then the $j$-th eigenvalue $\lambda_j$ is close to $\gamma_j$ in the sense that for any $\xi>1$ there is a constant $L$ such that \[\mathbb P \Big (\exists \, j : \; |\lambda_j-\gamma_j| \ge (\log N)^L \Big [ \min \big (\, j, N-j+1 \, \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{\xi} \big]} \] for $N$ large enough. (2) The proof of the {\it Dyson's conjecture} \cite{Dy} which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order $N^{-1}$. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large $N$ limit provided that the second moments of the two ensembles are identical.

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We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log$n$, with$n$being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the$negative$solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.

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We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri Poincar\'{e} Probab. Stat. 48 (1013) 107-133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.

Let each point of a homogeneous Poisson process in ℝ^{$d$}independently be equipped with a random number of stubs (half-edges) according to a given probability distribution$μ$on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution$μ$. Leaving aside degenerate cases, we prove that for any$μ$there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on$μ$for the absence and presence of infinite components.

The stochasticity of gene expression is manifested in the fluctuations of mRNA and protein copy numbers within a cell lineage over time. While data of this type can be obtained for many generations, most mathematical models are unsuitable to interpret such data since they assume non-growing cells. Here we develop a theoretical approach that quantitatively links the frequency content of lineage data to subcellular dynamics. We elucidate how the position, height, and width of the peaks in the power spectrum provide a distinctive fingerprint that encodes a wealth of information about mechanisms controlling transcription, translation, replication, degradation, bursting, promoter switching, cell cycle duration, cell division, gene dosage compensation, and cell size homeostasis. Predictions are confirmed by analysis of single-cell Escherichia coli data obtained using fluorescence microscopy. Furthermore, by matching the experimental and theoretical power spectra, we infer the temperature-dependent gene expression parameters, without the need of measurements relating fluorescence intensities to molecule numbers.