We give a survey of some of the principal results in percolation theory and first-passage percolation. No proofs are given. Open problems abound and are usually displayed with a number ($\ast$ P).

Using the invasion percolation process, we prove the following for Bernoulli percolation on $\mathbb{Z}^d (d > 2)$: (1) exponential decay of the truncated connectivity, $\tau'_{xy} \equiv P(x$ and $y$ belong to the same finite cluster$) \leq \exp(-m\|x - y\|)$; (2) infinite differentiability of $P_\infty(p)$, the infinite cluster density, and of $\chi'(p)$, the expected size of finite clusters, as functions of $p$, the density of occupied bonds; and (3) upper bounds on the cluster size distribution tail, $P_n \equiv P$(the cluster of the origin contains exactly $n$ bonds) $\leq \exp(-\lbrack c/\log n\rbrack n^{(d-1)/d})$. Such results (without the $\log n$ denominator in (3)) were previously known for $d = 2$ and $p > p_c$, the usual percolation threshold, or for $d > 2$ and $p$ close to 1. We establish these results for all $d > 2$ when $p$ is above a limit of "slab thresholds," conjectured to coincide with $p_c$.

We consider dynamical systems in $\mathbb{R}^d$ driven by a vector field $b(x) = - \nabla a(x)$, where $a$ is a double-well potential with some smoothness conditions. We show that these dynamical systems when subjected to a small random disturbance exhibit metastable behavior in the sense defined in [2]. More precisely, we prove that the process of moving averages along a path of such a system converges in law when properly normalized to a jump Markov process. The main tool for our analysis is the theory of Freidlin and Wentzell [7].

Let $(X, \mathscr{A})$ be a measurable space and $\mathscr{F}$ a class of measurable functions on $X. \mathscr{F}$ is called a universal Donsker class if for every probability measure $P$ on $\mathscr{A}$, the centered and normalized empirical measures $n^{1/2}(P_n - P)$ converge in law, with respect to uniform convergence over $\mathscr{F}$, to the limiting "Brownian bridge" process $G_P$. Then up to additive constants, $\mathscr{F}$ must be uniformly bounded. Several nonequivalent conditions are shown to imply the universal Donsker property. Some are connected with the Vapnik-Cervonenkis combinatorial condition on classes of sets, others with metric entropy. The implications between the various conditions are considered. Bounds are given for the metric entropy of convex hulls in Hilbert space.

Let $\mathscr{F}$ be a class of square integrable functions. We give necessary and sufficient random geometric conditions for the empirical process indexed by $\mathscr{F}$ to satisfy the CLT. These conditions roughly mean that the trace of $\mathscr{F}$ on a random sample is a small (for the $l^1$ norm) perturbation of a set which is nice for the $l^2$ norm.

Initially we consider "the" standard isonormal linear process $L$ on a Hilbert space $H$, and applying metric entropy methods obtain bounds for the probability that $\sup_CLx > \lambda, C \subset H$ and $\lambda$ large. Under the assumption that the entropy function of $C$ grows polynomially, we find bounds of the form $c\lambda^\alpha\exp(- \frac{1}{2}\lambda^2/\sigma^2)$, where $\sigma^2$ is the maximal variance of $L$. We use a notion of entropy finer than that usually employed and specifically suited to the nonstationary situation. As a result we obtain, in the nonstationary setting, more precise bounds than any in the literature. We then treat a number of examples in which the power $\alpha$ is identified. These include the distributions of the maxima of the rectangle indexed, pinned Brownian sheet on $\mathbb{R}^k$ for which $\alpha = 2(2k - 1)$, and the half plane indexed pinned sheet on $\mathbb{R}^2$ for which $\alpha = 2$.

It is shown that when the index $0 < \alpha < 2, \alpha \neq 1$, and the symmetry parameter $-1 \leq \beta \leq 1$ of a stable process $\{X(t); t \geq 0\}$ are such that $P\{X(1) > 0\} = l\alpha^{-1} - k$, where $l$ and $k$ are integers, Darling's integral can be evaluated. This leads to explicit formulas for a transform of the Laplace transform of $\sup_{0\leq t\leq 1}X(t)$ and the Wiener-Hopf factors of $\{X(t), t \geq 0\}$.

Let $S_n = X_1 + \cdots + X_n$ be the $n$th partial sum of an i.i.d. sequence of random variables. We describe the limiting behavior of \begin{equation*}\begin{split}T_n = \max_{1\leq i\leq n}(S_{i+\kappa(i)} - S_i), \\ U_n = \max_{0\leq i\leq n-k}(S_{i+k} - S_i), \\ W_n = \max_{0\leq i\leq n-k} \max_{1\leq j\leq k}(S_{i+j} - S_i) \\ \end{split}\end{equation*} and $V_n = \max_{0\leq i\leq n-k} \min_{1\leq j\leq k}(k/j)(S_{i+j} - S_i),$ for $k = \kappa(n) = \lbrack c \log n\rbrack$, and where $c > 0$ is a given constant. We assume that the random variables $X_i$ are centered and have a finite moment generating function in a right neighborhood of zero, and obtain among other results the full form of the Erdos-Renyi (1970) and Shepp (1964) theorems. Our conditions extend those of Deheuvels, Devroye and Lynch (1986) to cover a larger class of distributions.

In this note we establish the central limit theorem for $\rho$-mixing sequences under combinations of moment assumptions and $\rho$-mixing rates. These results contain the well-known Ibragimov theorems and answer a problem from recent work of Bradley.

A Gaussian central limit theorem for trimmed sums of $\Phi$-mixing Hilbert-space-valued random variables is obtained, and implications regarding Ibragimov's conjecture are examined.

An estimate in the multidimensional central limit theorem is obtained, which is used together with the Strassen-Dudley theorem to prove a strong approximation theorem for partial sums of independent, identically distributed $d$-dimensional random vectors. This theorem implies immediately multi-dimensional versions of the strong invariance principles of Strassen and Major as well as a new $d$-dimensional strong invariance principle which improves the known results for the 1-dimensional case. In particular, we are able to weaken the assumption in Major's strong invariance principle. At the same time, it is shown that the assumptions of our theorem are nearly necessary.

We present a general methodology for proving invariance principles for renewal processes, resulting in almost sure and probability inequality approximations. We show that our obtained rates are best possible in the i.i.d. case.

Let $(S_j)$ be a lattice random walk, i.e., $S_j = X_1 + \cdots + X_j$, where $X_1, X_2,\ldots$ are independent random variables with values in $\mathbb{Z}$ and common nondegenerate distribution $F$. Let $\{t_n\}$ be a nondecreasing sequence of positive integers, $t_n \leq n$, and $L^\ast_n = \max_{0\leq j\leq n-t_n}(L_{j+t_n} - L_j)$, where $L_n = \sum^n_{j=1}1_{\{0\}}(S_j)$, the number of times zero is visited by the random walk by time $n$. Assuming that the random walk is recurrent and satisfies a more general condition than being in the domain of attraction of a stable law of index $\alpha > 1$, the following results are obtained: (i) Constants $\beta_n$ are defined such that $\lim \sup L^\ast_n\beta^{-1}_n = 1$ a.s. (ii) If $\lim \sup nt^{-1}_n = \infty$, then constants $\gamma_n$ are defined such that $\lim \inf L^\ast_n\gamma^{-1}_n = 1$ a.s. If $\lim \sup nt^{-1}_n < \infty$, then $\lim \inf(L^\ast_n/\gamma'_n) = 0$ or $\infty$ for any choice of $\gamma'_n$ and a simple test is given to determine which is the case. (iii) If $\lim \log(nt^{-1}_n)/\log_2n = \infty$, then $\beta_n \sim \gamma_n$ and $\lim L^\ast_n\beta^{-1}_n = 1$ a.s. Also, the normalizers are found more explicitly in the domain of attraction case.

The usual Ito formula fails to apply for $r(X)$ when $r$ is a distance function and $X$ a Brownian motion on a general manifold, since $r$ fails to be differentiable on the cut-locus. It is shown that the discrepancy between the two sides of Ito's formula forms a monotonic random process (and hence is of locally bounded variation). In particular, $r(X)$ is a semimartingale.

Let $D$ be a domain in $\mathbb{R}^d$ and let $\Delta_1$ be the set of minimal points of the Martin boundary of $D$. For $x \in D$ and $z \in \Delta_1$, let $(X_t)$ under the law $P^{x; z}$ be Brownian motion in $D$, starting at $x$ and conditioned to converge to $z$. Let $\tau$ be the lifetime of $(X_t)$, so $X_{\tau-} = z P^{x; z}$ a.s. Let $q \in L^p(D)$ for some $p > d/2$. Under the assumption that $D$ is what we call rapidly exhaustible, which is essentially a very weak boundary smoothness condition, we show that if the quantity $E^{x;z}\big\{\exp\big\lbrack\int^\tau_0 q(X_s) ds\big\rbrack\big\}$ is finite for one $x \in D$ and one $z \in \Delta_1$, then this quantity is bounded on $D \times \Delta_1$. This result may be viewed as saying, in a fairly strong sense, that the amount of time $(X_t)$ spends in each part of $D$ does not depend very much on the minimal Martin boundary point $z$ to which $(X_t)$ is conditioned to converge.

By a well-known theorem of P. Levy, if $(X_t)$ is a standard Brownian motion on $\mathbb{R}$ with $X_0 = 0$ and if $H_t = \min_{u \leq t}X_u$, then $(Y_t) = (X_t - H_t)$ is Brownian motion with 0 as a reflecting lower boundary. More generally, if $X$ is allowed to have nonzero drift or a reflecting lower boundary at $A < 0$, then the process $Y = X - H$ is still a diffusion process. We prove the converse result: If $X$ is a diffusion on an interval $I \subset \mathbb{R}$ which contains 0 as an interior point, and if $(Y_t) = (X_t - H_t)$ is a time homogeneous strong Markov process (when $X_0 = 0$), then $X$ must be a Brownian motion on $I$ (with drift $\mu$, variance parameter $\sigma^2 > 0$, killing rate $c \geq 0$ and reflection at $\inf I$ in case $\inf I > -\infty$).

We find the moment generating function of the time at which Brownian motion exits from a region bounded by two nonconvergent straight lines, using the martingale stopping theorem.

A general framework is proposed for continuous time dynamic allocation models of a scarce resource among competing projects. The allocation model is formulated as a multi-armed bandit model and solved as a control problem of a multiparameter process. In contrast to discrete time bandits, where only one arm can be pulled at a time, the continuous time bandit must allow simultaneous pulls. The multiparameter approach allows a strong solution of diffusion-type bandits. Here the main problem is to define precisely how to switch among arms and the solution involves local times.

We consider infinite source collision detect random access schemes. For such schemes we establish that the number of packets successfully transmitted is infinite with probability 0 or 1 according as the arrival rate is greater than or less than a critical value.

Let $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ be independent sequences of i.i.d. $U(0, 1)$ random variables. We characterize completely those Borel functions $F$ on $\lbrack 0, 1\rbrack^2$ for which the strong law of large numbers and the maximal ergodic theorem hold for the doubly indexed family $(1/nm)\sum_{i \leq n, j \leq m}F(X_i, Y_j)$.

If $Y_n$ is 1 or 0 depending on whether the $n$th ball drawn in a Polya urn scheme is red or not, then the variables $Y_1, Y_2,\ldots$ are exchangeable. It is shown for a generalized class of urn models that no other scheme gives rise to exchangeable variables unless the $Y_n$ are either independent and identically distributed, or deterministic (that is, all of the $Y_n$'s have the same value with probability 1).

A prophet is a player with complete foresight; a gambler knows only the past and the present, but not the future. If each of them bets on differences of consecutive nonnegative random variables $X_i$ such that $E(X_i|X_{i - 1}) = EX_i$, the players multiplying their stakes by uniformly bounded variables, then the expected gain of the prophet is at most three times that of the gambler. The constant 3 is optimal.

The notion of adapted distribution of a stochastic process was introduced in a recent paper of Hoover and Keisler. Here we give a simple characterization of this notion in terms of filtration embeddability. This characterization allows us to show that for a local martingale $M$ for which some ordinary stochastic differential equation $X_t = \int^t_0f(s, X_s) dM_s$ admits sufficient nonuniqueness in law of the solutions $X$, the class of possible joint laws of $(M, X)$ determines the adapted law of $M$.