The metric $D_{\alpha}(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $\mathbb{R}^d$ , defined as the infimum of $(\sum_i |q_i - q_{i+1}|^{\alpha})^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $|·|$ denotes Euclidean distance) has nontrivial geodesics when $\alpha>1$. The cases $1< \alpha<\infty$ are the Euclidean first­passage percolation (FPP) models introduced earlier by the authors, while the geodesics in the case $\alpha = \infty$ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conjectures for these two situations. New results for $1 < \alpha < \infty$ (and any $d$) include inequalities on the fluctuation exponents for the metric $(\chi \le 1/2)$ and for the geodesics $(\xi \le 3/4)$ in strong enough versions to yield conclusions not yet obtained for lattice FPP: almost surely, every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic (from every $q$). For $d = 2$ and $2 \le \alpha < \infty$, further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces.

We consider a version of the Boolean (or Poisson blob) continuum percolation model where, at each point of a Poisson point process in the Euclidean plane with intensity $\lambda$, a copy of a given compact convex set $A$ with fixed rotation is placed. To each $A$ we associate a critical value $\lambda_c (A)$ which is the infimum of intensities $\lambda$ for which the occupied component contains an unbounded connected component. It is shown that $\min\{\lambda_c(A):A \text{ convex of area } a\}$ is attained if $A$ is any triangle of area $a$ and $\max\{\lambda_c(A):A \text{ convex of area } a\}$ is attained for some centrally symmetric convex set $A$ of area $a$.

It turns out that the key result, which is also of independent interest, is a strong version of the difference­body inequality for convex sets in the plane. In the plane, the difference­body inequality states that for any compact convex set $A, 4\mu (A) \le \mu (A \oplus \check{A}) \le 6\mu (A)$ with equality to the left iff $A$ is centrally symmetric and with equality to the right iff $A$ is a triangle. Here $\mu$ denotes area and $A \oplus \check{A}$ is the difference­body of $A$. We strengthen this to the following result: For any compact convex set $A$ there exist a centrally symmetric convex set $C$ and a triangle $T$ such that $\mu(C) = \mu(T) = \mu(A)$ and $C \oplus \check{C} \subseteq A \oplus \check{A} \subseteq T \oplus \check{T}$ with equality to the left iff $A$ is centrally symmetric and to the right iff $A$ is a triangle.

We consider the parabolic Anderson problem $\partial_t u = \kappa\Delta u + \xi u$ on $(0,\infty) \times \mathbb{Z}^d$ with random i.i.d. potential $\xi = (\xi(z))_{z\in \mathbb{Z}^d}$ and the initial condition $u(0,\cdot) \equiv 1$. Our main assumption is that $\mathrm{esssup} \xi(0)=0$. Depending on the thickness of the distribution $\mathrm{Prob} (\xi(0) \in \cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t, 0)$ and the almost­sure asymptotics of $u(t, 0)$ as $t \to \infty$ in terms of variational problems. As a by­product, we establish Lifshitz tails for the random Schrödinger operator $-\kappa \Delta - \xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.

Let $L_n$ be the length of the longest increasing subsequence of a random permutation of the numbers $1,\ldots, n$, for the uniform distribution on the set of permutations. Hammersley's interacting particle process, implicit in Hammersley (1972), has been used in Aldous and Diaconis (1995) to provide a “soft” hydrodynamical argument for proving that $\lim_{n \to \infty} EL_n / \sqrt{n} = 2$. We show in this note that the latter result is in fact an immediate consequence of properties of a random 2­dimensional signed measure, associated with Hammersley’s process.

Let $t > 0, K$ be a connected compact Lie group equipped with an $Ad_K$- invariant inner product on the Lie Algebra of $K$. Associated to this data are two measures $\mu^0_t$ and $\nu^0_t$ on $\mathscr{L}(K)$ – the space of continuous loops based at $e \in K$. The measure $\mu^0_t$ is pinned Wiener measure with “variance $t$” while the measure $\nu^0_t$ is a “heat kernel measure” on $\mathscr{L}(K)$. The measure $\mu^0_t$ is constructed using a $K$-valued Brownian motion while the measure $\nu^0_t$ is constructed using a $\mathscr{L}(K)$-valued Brownian motion. In this paper we show that $\nu^0_t$ is absolutely continuous with respect to $\mu^0_t$ and the Radon-Nikodym derivative $d\nu^0_t /d\mu^0_t$ is bounded.

We introduce in this article a class of transient random walks in a random environment on $\mathbb{Z}^d$. When $d\ge 2$, these walks are ballistic and we derive a law of large numbers, a central limit theorem and large-deviation estimates. In the so-called nestling situation, large deviations in the neighborhood of the segment $[0, v]$, $v$ being the limiting velocity, are critical. They are of special interest in view of their close connection with the presence of traps in the medium, that is, pockets where a certain spectral parameter takes atypically low values.

In this paper we develop a stochastic calculus with respect to a Gaussian process of the form $B_t = \int^t_0 K(t, s)\, dW_s$, where $W$ is a Wiener process and $K(t, s)$ is a square integrable kernel, using the techniques of the stochastic calculus of variations. We deduce change-of-variable formulas for the indefinite integrals and we study the approximation by Riemann sums.The particular case of the fractional Brownian motion is discussed.

We show that a strong mixing condition implies a Bakry-Bobkov-Ledoux inequality for a probability measure on infinite-dimensional space.

This paper aims to combine the central limit theorem with the limit theorems in extreme value theory through a parametrized class of limit theorems where the former ones appear as special cases. To this end the limit distributions of suitably centered and normalized $l_{cp(n)}$-norms of $n$-vectors of positive i.i.d. random variables are investigated. Here, $c$ is a positive constant and $p(n)$ is a sequence of positive numbers that is given intrinsically by the form of the upper tail behavior of the random variables. A family of limit distributions is obtained if $c$ runs over the positive real axis. The normal distribution and the extreme value distributions appear as the endpoints of these families, namely, for $c =0 +$ and $c = \infty$, respectively.

Let $D$ be the interior of a parabola in $\mathbb{R}^2$ and $\tau_D$ the first exit time of Brownian motion from $D$. We show $-\log P(\tau_D) >t)$ behaves like $t^{1 /3}$ as $t \to \infty$.

We present a probabilistic approach for the study of systems with exclusions in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with “duplicated-variables expansion,” used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques.

We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) $\mathscr{S}$ when the measures are indexed by another poset $\mathscr{A}$. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of $\mathscr{S}$ we also present conditions on the poset $\mathscr{A}$ that are necessary and sufficient for equivalence. When $\mathscr{A} = \mathscr{S}$ , the condition that the cover graph of $\mathscr{S}$ have no cycles is necessary and sufficient for equivalence. This case arises in comparing applicability of the perfect sampling algorithms of Propp and Wilson and the first author of the present paper.

Let $G \subset \mathbb{R}^k$ be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces $\{G_i, i =1,\ldots, N\}$, where $n_i$ and $d_i$ denote the inward normal and direction of constraint associated with $G_i$, respectively. Stability properties of a class of diffusion processes, constrained to take values in $G$, are studied under the assumption that the Skorokhod problem defined by the data $\{(n_i,d_i),i = 1,\ldots,N\}$ is well posed and the Skorokhod map is Lipschitz continuous. Explicit conditions on the drift coefficient, $b(\cdot)$, of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure. Define $$\mathscr{C}\doteq \left\{ - \sum^{N}_{i=1} a_i d_i;a_i \ge 0, i \in \{1,\ldots,N\} \right\}.$$ Then the key condition for stability is that there exists $\delta \in (0,\infty)$ and a bounded subset $A$ of $G$ such that for all $x \in G\setminus A, b(x) \in \mathscr{C}$ and $\mathrm{dist} (b(x),\partial\mathscr{C}) \ge \delta$, where $\partial\mathscr{C}$denotes the boundary of $\mathscr{C}$.