In this paper we look at the pinning of a directed polymer by a one-dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. Theory Related Fields 148 (2010) 403–456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. These variational formulas are used to obtain a necessary and sufficient criterion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature. Subsequently, upper and lower bounds are derived for the inverse critical temperature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be necessary as well, a problem that we will address in a forthcoming paper.

Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach, that is, flexible enough to be extended to other models of random polymers with disorder.

The Marcus–Lushnikov process is a simple mean field model of coagulating particles that converges to the homogeneous Smoluchowski equation in the large mass limit. If the coagulation rates grow sufficiently fast as the size of particles get large, giant particles emerge in finite time. This is known as gelation, and such particles are known as gels. Gelation comes in different flavors: simple, instantaneous and complete. In the case of an instantaneous gelation, giant particles are formed in a very short time. If all particles coagulate to form a single particle in a time interval that stays bounded as total mass gets large, then we have a complete gelation. In this article, we describe conditions which guarantee any of the three possible gelations with explicit bounds on the size of gels and the time of their creations.

In this paper, we study the stability and convergence of some general quadratic semimartingales. Motivated by financial applications, we study simultaneously the semimartingale and its opposite. Their characterization and integrability properties are obtained through some useful exponential submartingale inequalities. Then, a general stability result, including the strong convergence of the martingale parts in various spaces ranging from $\mathbb{H}^{1}$ to BMO, is derived under some mild integrability condition on the exponential of the terminal value of the semimartingale. This can be applied in particular to BSDE-like semimartingales.

This strong convergence result is then used to prove the existence of solutions of general quadratic BSDEs under minimal exponential integrability assumptions, relying on a regularization in both linear-quadratic growth of the quadratic coefficient itself. On the contrary to most of the existing literature, it does not involve the seminal result of Kobylanski [Ann. Probab. 28 (2010) 558–602] on bounded solutions.

Consider a branching random walk on $\mathbb{R}$, with offspring distribution $Z$ and nonnegative displacement distribution $W$. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution $Z$ is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs $(Z,W)$ for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur.

We also investigate, under additional smoothness assumptions, the behavior of $M_{n}$, the position of the particle in generation $n$ closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_{n}=\infty$).

Sinai’s random walk in random environment shows interesting patterns on the exponential time scale. We characterize the patterns that appear on infinitely many time scales after appropriate rescaling (a functional law of iterated logarithm). The curious rate function captures the difference between one-sided and two-sided behavior.

Let $M$ be a compact Riemannian homogeneous space (e.g., a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation $\mathbf{D}_{t}\partial_{t}u=\sum_{k=1}^{d}\mathbf{D}_{x_{k}}\partial_{x_{k}}u+f_{u}(Du)+g_{u}(Du)\dot{W}$ in any dimension $d\ge1$, where $f$ and $g$ are continuous multilinear maps, and $W$ is a spatially homogeneous Wiener process on $\mathbb{R}^{d}$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.

Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.

We consider the Bernoulli bond percolation process $\mathbb{P} _{p,p'}$ on the nearest-neighbor edges of $\mathbb{Z} ^{d}$, which are open independently with probability $p<p_{c}$, except for those lying on the first coordinate axis, for which this probability is $p'$. Define

\[\xi_{p,p'}:=-\lim_{n\to\infty}n^{-1}\log\mathbb{P} _{p,p'}(0\leftrightarrow n\mathbf{e} _{1})\]

and $\xi_{p}:=\xi_{p,p}$. We show that there exists $p_{c}'=p_{c}'(p,d)$ such that $\xi_{p,p'}=\xi_{p}$ if $p'<p_{c}'$ and $\xi_{p,p'}<\xi_{p}$ if $p'>p_{c}'$. Moreover, $p_{c}'(p,2)=p_{c}'(p,3)=p$, and $p_{c}'(p,d)>p$ for $d\geq 4$. We also analyze the behavior of $\xi_{p}-\xi_{p,p'}$ as $p'\downarrow p_{c}'$ in dimensions $d=2,3$. Finally, we prove that when $p'>p_{c}'$, the following purely exponential asymptotics holds:

\[\mathbb{P} _{p,p'}(0\leftrightarrow n\mathbf{e} _{1})=\psi_{d}e^{-\xi_{p,p'}n}\bigl(1+o(1)\bigr)\]

for some constant $\psi_{d}=\psi_{d}(p,p')$, uniformly for large values of $n$. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and don’t rely on exact computations.

In this paper we study the supremum functional $M_{t}=\sup_{0\le s\le t}X_{s}$, where $X_{t}$, $t\ge0$, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_{t}$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_{t}$ if the Lévy–Khintchin exponent of the process increases on $(0,\infty)$.

We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of “global stability at a finite distance”: roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the line. For instance, we show that under a suitable change of the coordinates, the drift of every point becomes zero provided that the action is minimal. As a byproduct, we recover the fact that every finitely generated group of homeomorphisms of the real line is topologically conjugate to a group of (globally) Lipschitz homeomorphisms. Moreover, we show that such a conjugacy may be chosen in such a way that the displacement of each element is uniformly bounded.

Let $S$ be a Polish space and $(X_{n}:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_{n}(\cdot)=P(X_{n+1}\in\cdot\mid X_{1},\ldots,X_{n})$ be the predictive measure and $\alpha$ a random probability measure on $S$ such that $\alpha_{n}\stackrel{\mathrm{weak}}{\longrightarrow}\alpha$ a.s. Two (related) problems are addressed. One is to give conditions for $\alpha\ll\lambda$ a.s., where $\lambda$ is a (nonrandom) $\sigma$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal{L}(X_{1},\ldots,X_{n})$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alpha_{n}-\alpha\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert\cdot\Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_{n})$ is conditionally identically distributed, in the sense of [Ann. Probab. 32 (2004) 2029–2052].

For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form “no eigenvalues outside the support of the limiting eigenvalue distribution.” We build on ideas of Haagerup–Schultz–Thorbjørnsen on the one hand and Bai–Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincaré-type inequalities, we use a variety of matrix identities and $L^{p}$ estimates. The Schwinger–Dyson equation controls much of the analysis.

We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp–Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cramér theorem.

We consider a nonlinear stochastic heat equation $\partial_{t}u=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space–time white noise and $\sigma:\mathbf{R} \to\mathbf{R} $ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_{0}$: under suitable conditions on $u_{0}$ and $\sigma$, $\sup_{x\in\mathbf{R} }u_{t}(x)$ is a.s. finite when $u_{0}$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_{t}(x)/({\log}|x|)^{1/6}>0$ when $u_{0}$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.

The intent of this paper is to describe the large scale asymptotic geometry of iteration stable (STIT) tessellations in $\mathbb{R}^{d}$, which form a rather new, rich and flexible class of random tessellations considered in stochastic geometry. For this purpose, martingale tools are combined with second-order formulas proved earlier to establish limit theorems for STIT tessellations. More precisely, a Gaussian functional central limit theorem for the surface increment process induced a by STIT tessellation relative to an initial time moment is shown. As second main result, a central limit theorem for the total edge length/facet surface is obtained, with a normal limit distribution in the planar case and, most interestingly, with a nonnormal limit showing up in all higher space dimensions.

We consider the ensemble of adjacency matrices of Erdős–Rényi random graphs, that is, graphs on $N$ vertices where every edge is chosen independently and with probability $p\equiv p(N)$. We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as $pN\to\infty$ (with a speed at least logarithmic in $N$), the density of eigenvalues of the Erdős–Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than $N^{-1}$ (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the $\ell^{\infty}$-norms of the $\ell^{2}$-normalized eigenvectors are at most of order $N^{-1/2}$ with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős–Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that $pN\gg N^{2/3}$.

The Tanaka equation $dX_{t}=\operatorname{sign}(X_{t})\,dB_{t}$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion $B$, then the solution of the obtained equation is pathwise unique.