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For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg and Hoffman (preprint, 2016), and others. However the main results do not cover geodesics in the direction of corners of the limit shape $\mathcal{B}$, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics. Consequently, the set of points of $\partial \mathcal{B}$ which are in the direction of some geodesic does not have all of $\mathcal{B}$ as its convex hull., 26 pages, 4 figures

The availability of suppositories in Hungary, especially in clinical pharmacy practice, is usually provided by extemporaneous preparations. Due to the known advantages of rectal drug administration, its benefits are frequently utilized in pediatrics. However, errors during the extemporaneous manufacturing process can lead to non-homogenous drug distribution within the dosage units. To determine the root cause of these errors and provide corrective actions, we studied suppository samples prepared with exactly known errors using both cerimetric titration and HPLC technique. Our results show that the most frequent technological error occurs when the pharmacist fails to use the correct displacement factor in the calculations which could lead to a 4.6% increase/decrease in the assay in individual dosage units. The second most important source of error can occur when the molding excess is calculated solely for the suppository base. This can further dilute the final suppository drug concentration causing the assay to be as low as 80%. As a conclusion we emphasize that the application of predetermined displacement factors in calculations for the formulation of suppositories is highly important, which enables the pharmacist to produce a final product containing exactly the determined dose of an active substance despite the different densities of the components.

Background. Skin infections occur commonly and often present therapeutic challenges to practitioners due to the growing concerns regarding multidrug-resistant bacterial, viral, and fungal strains. The antimicrobial properties of zinc sulfate and copper sulfate are well known and have been investigated for many years. However, the synergistic activity between these two metal ions as antimicrobial ingredients has not been evaluated in topical formulations. Objective. The aims of the present study were to (1) formulate topical creams and gels containing zinc and copper alone or in combination and (2) evaluate the in vitro antibacterial activity of these metal ions in the formulations. Method. Formulation of the gels and creams was followed by evaluating their organoleptic characteristics, physicochemical properties, and in vitro antibacterial activity against Escherichia coli and Staphylococcus aureus. Results. Zinc sulfate and copper sulfate had a strong synergistic antibacterial activity in the creams and gels. The minimum effective concentration was found to be 3 w/w% for both active ingredients against the two tested microorganisms. Conclusions. This study evaluated and confirmed the synergistic in vitro antibacterial effect of copper sulfate and zinc sulfate in a cream and two gels.

Organogels have emerged as an alternative carrier for small and macromolecules via transdermal, oral, rectal and ophthalmic routes. Pluronic lecithin organogels (PLO gels) are lecithin-based organogels widely used in compounding pharmacies as a vehicle for enhancing the transdermal permeability of many therapeutic drugs. However, the scientific and systematic evidence in support of how well PLO gels help in transdermal delivery is scanty. Recently, some clinical studies have reported nearly complete lack of bioavailability of certain topically administered drugs from PLO gels. The present review aims at summarizing gels and organogels, with a focus on the use of PLO gels in transdermal drug delivery. A special emphasis is placed on controversies looming over the use of PLO gels as a delivery platform for drugs via transdermal route. This article is open to POST-PUBLICATION REVIEW. Registered readers (see “For Readers”) may comment by clicking on ABSTRACT on the issue’s contents page.

We study the intersection of two independent renewal processes, $\rho=\tau\cap\sigma$. Assuming that $\mathbf{P}(\tau_1 = n ) = \varphi(n)\, n^{-(1+\alpha)}$ and $\mathbf{P}(\sigma_1 = n ) = \tilde\varphi(n)\, n^{-(1+ \tilde\alpha)} $ for some $\alpha,\tilde \alpha \geq 0$ and some slowly varying $\varphi,\tilde\varphi$, we give the asymptotic behavior first of $\mathbf{P}(\rho_1>n)$ (which is straightforward except in the case of $\min(\alpha,\tilde\alpha)=1$) and then of $\mathbf{P}(\rho_1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities $\mathbf{P}(\rho_1=n)$ while knowing asymptotically the renewal mass function $\mathbf{P}(n\in\rho)=\mathbf{P}(n\in\tau)\mathbf{P}(n\in\sigma)$. Our results can be used to bound coupling-related quantities, specifically the increments $|\mathbf{P}(n\in\tau)-\mathbf{P}(n-1\in\tau)|$ of the renewal mass function., Comment: 1 figure, 21 pages

We study the solid-on-solid interface model above a horizontal wall in three dimensional space, with an attractive interaction when the interface is in contact with the wall, at low temperatures. There is no bulk external field. The system presents a sequence of layering transitions, whose levels increase with the temperature, before reaching the wetting transition., Comment: 61 pages, 6 figures. Miscellaneous corrections and changes, primarily in Section 4. Figure 5 added.

We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c}\phi(n)$ with $c \geq 1$ and $\phi$ slowly varying. Comparing to the corresponding annealed system, in which the $V_n$ are effectively replaced by a constant, it is known that the quenched and annealed critical points differ at all temperatures for $3/22$, but only at low temperatures for $c3/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$., Comment: 33 pages

We study i.i.d. sums $\tau_k$ of nonnegative variables with index $0$: this means $\mathbf{P}(\tau_1=n) = \varphi(n) n^{-1}$, with $\varphi(\cdot)$ slowly varying, so that $\mathbf{E}(\tau_1^\varepsilon)=\infty$ for all $\varepsilon>0$. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of $\mathbf{P}(\tau_k=n)$ when $n$ is at least the typical length of $\tau_k$. A recent renewal theorem by Nagaev [21] is an immediate consequence: $\mathbf{P}(n\in\tau) \sim \mathbf{P}(\tau_1=n)/\mathbf{P}(\tau_1 > n)^2$ as $n\to\infty$. If instead we only assume regular variation of $\mathbf{P}(n\in\tau)$ and slow variation of $U_n:= \sum_{k=0}^n \mathbf{P}(k\in\tau)$, we obtain a similar equivalence but with $\mathbf{P}(\tau_1=n)$ replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of $\mathbf{P}(\tau_k \leq n)$ when $n$ is much smaller than the typical length of $\tau_k$., Comment: 19 pages. We are grateful to V. Wachtel for bringing the result of Nagaev to our attention, and to an anonymous referee for pointing out a shorter proof of Theorem 1.3. Several changes were made accordingly

We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $\sigma$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $\sigma$ have infinite mean. The "polymer" -- of length $\sigma_N$ -- is given by another renewal $\tau$, whose law is modified by the Boltzmann weight $\exp(\beta\sum_{n=1}^N \mathbf{1}_{\{\sigma_n\in\tau\}})$. Our assumption is that $\tau$ and $\sigma$ have gap distributions with power-law-decay exponents $1+\alpha$ and $1+\tilde \alpha$ respectively, with $\alpha\geq 0,\tilde \alpha>0$. There is a localization phase transition: above a critical value $\beta_c$ the free energy is positive, meaning that $\tau$ is \emph{pinned} on the quenched renewal $\sigma$. We consider the question of relevance of the disorder, that is to know when $\beta_c$ differs from its annealed counterpart $\beta_c^{\rm ann}$. We show that $\beta_c=\beta_c^{\rm ann}$ whenever $ \alpha+\tilde \alpha \geq 1$, and $\beta_c=0$ if and only if the renewal $\tau\cap\sigma$ is recurrent. On the other hand, we show $\beta_c>\beta_c^{\rm ann}$ when $ \alpha+\frac32\, \tilde \alpha \beta_c^{\rm ann}$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($\sigma_N=\tau_N$), and one in which the polymer length is $\tau_N$ rather than $\sigma_N$. In both cases we show the critical point is the same as in the original model, at least when $ \alpha>0$., Comment: 51 pages, 1 figure

We continue our study of colligative properties of solutions initiated in math-ph/0407034. We focus on the situations where, in a system of linear size $L$, the concentration and the chemical potential scale like $c=\xi/L$ and $h=b/L$, respectively. We find that there exists a critical value $\xit$ such that no phase separation occurs for $\xi\le\xit$ while, for $\xi>\xit$, the two phases of the solvent coexist for an interval of values of $b$. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when $b$ reaches a critical value. For certain values of system parameters, under ``frozen'' boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing $b$ and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezing-point depression is in fact a surface phenomenon., Comment: 27 pages, 1 fig; see also math-ph/0407034 (both to appear in JSP)

We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least $l^2$, for large $l$. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large $l$ this maximum is at least of order $l^{1/3}(\log l)^{-2/3}$. This complements the upper bound of order $l^{1/3}(\log l)^{2/3}$ known for the average local roughness. The exponent 1/3 on $l$ here is in keeping with predictions from the physics literature for interfaces in two dimensions., 28 pages, 1 figure (.eps file). See also http://math.usc.edu/~alexandr/

We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $\beta$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $\beta$) of the annealed critical point $u_c^a=0$. For the case $u=0$, we show that the difference between quenched and annealed free energies is of order $\beta^4$ as $\beta\to 0$, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions., Comment: 22 pages. Changes to Proposition 3.8 make Proposition 4.1 unnecessary; minor corrections made

Four non-polymeric plasticizers, propylene glycol, diethyl phthalate, triacetin, and glycerin have been subjected to rising temperature thermogravimetry for kinetic analysis and vaporization-based thermal stability evaluation. Since volatile loss of a substance is a function of its vapor pressure, the thermal stability of these plasticizers has been analyzed by generating vapor pressure curves using the Antoine and Langmuir equations. Unknown Antoine constants for the sample compounds, triacetin and glycerin have been derived by subjecting the vapor pressure curves to nonlinear regression. For the first time, the entire process of obtaining the unknown Antoine constants through thermogravimetry has been validated by developing an approach called the 'double reference method.' Based on this method, it has been possible to show that this technique is accurate even for structurally diverse compounds. Kinetic analysis on the volatilization of compounds revealed a predominant zero order process. The activation energy values for vaporization of propylene glycol, diethyl phthalate, triacetin, and glycerin, as deduced from the Arrhenius plots, have been determined to be 55.80, 66.45, 65.12, and 67.54 kJ/mol, respectively. The enthalpies of vaporization of the compounds have been determined from the Clausius-Clapeyron plots. Rising temperature thermogravimetry coupled with nonlinear regression analysis has been shown to be an effective and rapid technique for accurately predicting the vapor pressure behavior and thermal stability evaluation of volatile compounds.

We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense "tight in probability" as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer., Published in at http://dx.doi.org/10.1214/13-AAP930 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

For a family of bond percolation models on Z^{2} that includes the Fortuin-Kasteleyn random cluster model, we consider properties of the ``droplet'' that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Koteck\'y and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the ``local roughness'') is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l^{1/3}(\log l)^{2/3}) in probability, where l = \sqrt{A} is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of ``bottlenecks,'' which are a form of self-approach by the droplet boundary, down to scale \log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present., Comment: 72 pages

Using yeast forward and reverse two-hybrid analyses, we have discovered that the replication terminator protein Tus of Escherichia coli physically interacts with DnaB helicase in vivo . We have confirmed this protein–protein interaction in vitro . We show further that replication termination involves protein–protein interaction between Tus and DnaB at a critical region of Tus protein, called the L1 loop. Several mutations located in the L1 loop region not only reduced the protein–protein interaction but also eliminated or reduced the ability of the mutant forms of Tus to arrest DnaB at a Ter site. At least one mutation, E49K, significantly reduced Tus–DnaB interaction and almost completely eliminated the contrahelicase activity of Tus protein in vitro without significantly reducing the affinity of the mutant form of Tus for Ter DNA, in comparison with the wild-type protein. The results, considered along with the crystal structure of Tus– Ter complex, not only elucidate further the mechanism of helicase arrest but also explain the molecular basis of polarity of replication fork arrest at Ter sites.

For spin-systems with an internal symmetry, we provide sufficient conditions for unicity of the Gibbs state and/or complete analyticity by comparison to random cluster models.

Given a finite collection of strings of letters from a fixed alphabet, it is of interest, in the contexts of data compression and DNA sequencing, to find the length of the shortest string which contains each of the given strings as a consecutive substring. In order to analyze the average behavior of the optimal superstring length, substrings of specified lengths are considered with the letters selected independently at random. An asymptotic expression is obtained for the savings from compression, i.e. the difference between the uncompressed (concatenated) length and the optimal superstring length.

We consider directed random polymers in $(d+1)$ dimensions with nearly gamma i.i.d. disorder. We study the partition function $Z_{N,\omega}$ and establish exponential concentration of $\log Z_{N,\omega}$ about its mean on the subgaussian scale $\sqrt{N/\log N}$ . This is used to show that $\mathbb{E}[ \log Z_{N,\omega}]$ differs from $N$ times the free energy by an amount which is also subgaussian (i.e. $o(\sqrt{N})$), specifically $O(\sqrt{\frac{N}{\log N}}\log \log N)$., Comment: Minor changes. Appears in Electronic Journal of Probability, 18, 2013, no. 5

We consider a simple random walk W_i in 1 or 2 dimensions, in which the walker may choose to stand still for a limited time. The time horizon is n, the maximum consecutive time steps which can be spent standing still is m_n and the goal is to maximize P(W_n=0). We show that for dimension 1, if m_n grows faster than (\log n)^{2+\gamma} for some \gamma>0, there is a strategy for each n such that P(W_n = 0) approaches 1. For dimension 2, if m_n grows faster than a positive power of n then there are strategies keeping P(W_n=0) bounded away from 0., Comment: 7 pages

Polymeric nanosuspension was prepared from an inert polymer resin (Eudragit® RL100) with the aim of improving the availability of sulfacetamide at the intraocular level to combat bacterial infections.Nanosuspensions were prepared by the solvent displacement method using acetone and Pluronic® F108 solution. Drug to polymer ratio was selected as formulation variable. Characterization of the nanosupension was performed by measuring particle size, zeta potential, Fourier Transform infrared spectra (FTIR), Differential Scanning Calorimetry (DSC), Powder X-Ray Diffraction (PXRD), drug entrapment efficiency and in vitro release. In addition, freeze drying, redispersibility and short term stability study at room temperature and at 4(0)C were performed.Spherical, uniform particles (size below 500 nm) with positive zeta potential were obtained. No significant chemical interactions between drug and polymer were observed in the solid state characterization of the freeze dried nanosuspension (FDN). Drug entrapment efficiency of the selected batch was increased by changing the pH of the external phase and addition of polymethyl methacrylate in the formulation. The prepared nanosuspension exhibited good stability after storage at room temperature and at 4(0)C. Sucrose and Mannitol were used as cryoprotectants and exhibited good water redispersibility of the FDN.The results indicate that the formulation of sulfacetamide in Eudragit® RL100 nanosuspension could be utilized as potential delivery system for treating ocular bacterial infections.

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -\delta/2x + o(1/x) with \delta > -1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}\phi(n), where c = (3+\delta)/2 and \phi is a slowly varying function given explicitly in terms of the o(1/x) terms., Comment: 44 pages. Numerous small corrections and clarifications. References added

International audience; We consider the three-dimensional Ising model in a half-space with a boundary field (no bulk field). We compute the low-temperature expansion of layering transition lines.

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length $n$ is given by $n^{-c}\phi(n)$ for some $13/2$, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point--the size of this neighborhood scales as $\beta^{1/(2c-3)}$ where $\beta$ is the inverse temperature. For $c0$, for sufficiently high temperature the quenched and annealed curves are within a factor of $1-\epsilon$ for all $u$ near the critical point; in particular the quenched and annealed critical points are equal. For $c=3/2$ the regime depends on the slowly varying function $\phi$., Comment: 31 pages

We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. Disorder is introduced by, for example, having the interaction vary from one monomer to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a critical value of $u$ above which the polymer is pinned, placing a positive fraction of its monomers at 0 with high probability. This critical point may differ for the quenched, annealed and deterministic cases. We show that self-averaging occurs, meaning that the quenched free energy and critical point are nonrandom, off a null set. We evaluate the critical point for a deterministic interaction ($u$ without added randomness) and establish our main result that the critical point in the quenched case is strictly smaller. We show that, for every fixed $u\in\mathbb{R}$, pinning occurs at sufficiently low temperatures. If the excursion length distribution has polynomial tails and the interaction does not have a finite exponential moment, then pinning occurs for all $u\in\mathbb{R}$ at arbitrary temperature. Our results apply to other mathematically similar situations as well, such as a directed polymer that interacts with a random potential located in a one-dimensional defect, or an interface in two dimensions interacting with a random potential along a wall., Comment: Published at http://dx.doi.org/10.1214/105051606000000015 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

An infinite-volume mixing or exponential-decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions, but in some circumstances where such properties hold, propagation can nonetheless occur in finite volumes endowed with boundary conditions. We establish the absense [sic] of such propagation, particularly in two dimensions in finite volumes which are simply connected, under a variety of conditions, mainly for the Potts model and the Fortuin--Kasteleyn (FK) random cluster model, allowing external fields. For example, for the FK model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes, uniformly over all such volumes and all boundary conditions, and implies a strong mixing property for such volumes with certain types of boundary conditions. For the Potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes, which includes exponential decay of correlations in simply connected finite volumes, uniformly over all such volumes and all boundary conditions.

Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based model of a solvent-solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing point depression. The limit of infinitesimal concentrations is described in a subsequent paper., Comment: 28 pages, 1 fig; see also math-ph/0407035 (both to appear in JSP)

Consider a translation-invariant bond percolation model on the integer lattice which has exponential decay of connectivities, that is, the probability of a connection $0 \leftrightarrow x$ by a path of open bonds decreases like $\exp\{-m(\theta)|x|\}$ for some positive constant $m(\theta)$ which may depend on the direction $\theta = x/|x|$. In two and three dimensions, it is shown that if the model has an appropriate mixing property and satisfies a special case of the FKG property, then there is at most a power-law correction to the exponential decay—there exist $A$ and $C$ such that $\exp\{-m(\theta)|x|\} \ge P(0 \leftrightarrow x) \ge A|x|^{-C} \exp\{-m(\theta)|x|\}$ for all nonzero $x$ . In four or more dimensions, a similar bound holds with $|x|^{-C}$ replaced by $\exp\{-C(\log |x|)^2\}$. In particular the power-law lower bound holds for the Fortuin-Kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially, since the mixing property is known to hold in that case. Consequently a similar bound holds for correlations in the Potts model at supercritical temperatures.

For a nonnegative subadditive function $h$ on $\mathbb{Z}^d$, with limiting approximation $g(x) = \lim_n h(nx)/n$, it is of interest to obtain bounds on the discrepancy between $g(x)$ and $h(x)$, typically of order $|x|^{\nu}$ with $\nu < 1$. For certain subadditive $h(x)$, particularly those which are expectations associated with optimal random paths from 0 to $x$, in a somewhat standardized way a more natural and seemingly weaker property can be established: every $x$ is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound. We show that this convex-hull property implies the desired bound for all $x$. Applications include rates of convergence in limiting-shape results for first-passage percolation (standard and oriented) and longest common subsequences and bounds on the error in the exponential-decay approximation to the off-axis connectivity function for subcritical Bernoulli bond percolation on the integer lattice.

Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the "treelike" random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line "winding through every region of the plane." If the random field has the FKG property, then only bounded level lines are possible. For stationary $C^2$ Gaussian random fields with covariance function decaying to 0 at $\infty$, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.

We prove a central limit theorem for the length of the minimal spanning tree of the set of sites of a Poisson process of intensity $\lambda$ in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the main difficulty is the dependency between the contributions to this length from different regions of $[0, 1]^2$; a percolation-theoretic result on circuits surrounding a fixed site can be used to control this dependency. We prove such a result via a continuum percolation version of the Russo-Seymour-Welsh theorem for occupied crossings of a rectangle. This RSW theorem also yields a variety of results for two-dimensional fixed-radius continuum percolation already well known for lattice models, including a finite-box criterion for percolation and absence of percolation at the critical point.

Ted Harris was born January 11, 1919, in Philadelphia, Pennsylvania. He grew up in Dallas, Texas, attended Southern Methodist University for two years and completed his undergraduate studies and some graduate work at the University of Texas at Austin. During World War II he served as a weather officer in England in the Army Air Force. He received his Ph.D. in 1947 from Princeton under Sam Wilks. From 1947 to 1966 he was a member of the mathematics department at The Rand Corporation in Santa Monica, California; he headed the department from 1959 to 1965. From 1966 to 1989 he was Professor of Mathematics and Electrical Engineering at the University of Southern California. Since 1989 he has been Professor Emeritus and Lecturer. In 1988 he was elected to the National Academy of Sciences, and in 1989 he received an honorary doctorate from Chalmers Institute of Technology, Sweden. He received an Albert S. Raubenheimer Distinguished Faculty Award in 1985 and a Distinguished Emeritus Award in 1990 from USC.

In processes such as invasion percolation and certain models of continuum percolation, in which a possibly random labelf(b) is attached to each bondb of a possibly random graph, percolation models for various values of a parameterr are naturally coupled: one can define a bondb to be occupied at levelr iff(b)≦r. If the labeled graph is stationary, then under the mild additional assumption of positive finite energy, a result of Gandolfi, Keane, and Newman ensures that, in lattice models, for each fixedr at which percolation occurs, the infinite cluster is unique a.s. Analogous results exist for certain continuum models. A unifying framework is given for such fixed-r results, and it is shown that if the site density is finite and the labeled graph has positive finite energy, then with probability one, uniqueness holds simultaneously for all values ofr. An example is given to show that when the site density is infinite, positive finite energy does not ensure uniqueness, even for fixedr. In addition, with finite site density but without positive finite energy, one can have fixed-r uniqueness a.s. for eachr, yet not have simultaneous uniqueness.

The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value $f(b)$ attached to each bond $b$. Of particular interest are stationary random graphs; examples include a lattice with iid uniform values $f(b)$ and the Voronoi or complete graph on the sites of a Poisson process, with $f(b)$ the length of $b$. The corresponding percolation models are Bernoulli bond percolation and the "lily pad" model of continuum percolation, respectively. It is shown that under a mild "simultaneous uniqueness" hypothesis, with at most one exception, each tree in the forest has one topological end, that is, has no doubly infinite paths. If there is a tree in the forest, necessarily unique, with two topological ends, it must contain all sites of an infinite cluster at the critical point in the corresponding percolation model. Trees with zero, or three or more, topological ends are not possible. Applications to invasion percolation are given. If all trees are one-ended, there is a unique optimal (locally minimax for $f$) path to infinity from each site.

Given two i.i.d. sequences of $n$ letters from a finite alphabet, one can consider the length $L_n$ of the longest sequence which is a subsequence of both the given sequences. It is known that $EL_n$ grows like $\gamma n$ for some $\gamma \in \lbrack 0, 1\rbrack$. Here it is shown that $\gamma n \geq EL_n \geq \gamma n - C(n \log n)^{1/2}$ for an explicit numerical constant $C$ which does not depend on the distribution of the letters. In simulations with $n = 100,000, EL_n/n$ can be determined from $k$ such trials with 95% confidence to within $0.0055/\sqrt k$, and the results here show that $\gamma$ can then be determined with 95% confidence to within $0.0225 + 0.0055/\sqrt k$, for an arbitrary letter distribution.

Functionals $L$ on finite subsets $A$ of $\mathbb{R}^d$ are considered for which the value is the minimum total edge length among a class of graphs with vertex set equal to, or in some cases containing, $A$. Examples include minimal spanning trees, the traveling salesman problem, minimal matching and Steiner trees. Beardwood, Halton and Hammersley, and later Steele, have shown essentially that for $\{X_1, \ldots, X_n\}$ a uniform i.i.d. sample from $\lbrack 0,1 \rbrack^d, EL(\{X_1, \ldots, X_n\})/n^{(d-1)/d}$ converges to a finite constant. Here we bound the rate of this convergence, proving a conjecture of Beardwood, Halton and Hammersley.

A variation is given of the van den Berg-Kesten inequality on the probability of disjoint occurrence of events enabling it to apply to random variables, rather than just to events, associated with various subsets of an index set. This is used to establish superadditivity of a certain family of generating functions associated with first-passage percolation. This leads to improved estimates for the rates of convergence of the expected values of certain passage times.

Given a finite alphabet, there is an inductive method for constructing a stationary measure on doubly infinite words from this alphabet. This construction can be randomized; the main focus here is on a particular uniform randomization which intuitively corresponds to the idea of choosing a generic stationary process. It is shown that with probability 1, the random stationary process has zero entropy and gives positive probability to every periodic infinite word.

The probability $P\lbrack 0 \leftrightarrow x \rbrack$ of connection of 0 to $x$ by a path of occupied bonds for Bernoulli percolation at density $p$ below the critical point is known to decay exponentially for each direction $x \in \mathbb{Z}^d$, in that $P\lbrack 0 \leftrightarrow nx \rbrack \approx e^{-n\sigma g(x)}$ as $n \rightarrow \infty$ for some $\sigma > 0$ and $g(x)$ of order $\|x\|$. This approximation is also an upper bound: $P\lbrack 0 \leftrightarrow x \rbrack \leq e^{-\sigma g(x)}$ for all $x$. Here a complementary power-law lower bound is established for $d = 2$ and 3: $P\lbrack 0 \leftrightarrow x \rbrack \geq c\|x\|^{-r} e^{-\sigma g(x)}$ for some $r = r(d)$ and $c = c(p,d)$.

Sufficient conditions are found for the weak convergence of a weighted empirical process { ( ν n (C) q(P(C)) ) 1 [P(C) ⪰ λ n ] : C ∈ C }, indexed by a class C of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q , determine the fastest rate at which one may allow λ n → 0.

Given a class $\mathscr{A}$ of subsets of $\lbrack 0, 1\rbrack^d$ and an array $\{X_j: \mathbf{j} \in \mathbb{Z}^d_+\}$ of independent identically distributed random variables with $EX_j = 0, EX^2_j = 1$, the (unsmoothed) partial-sum process $S_n$ is given by $S_n(A) := n^{-d/2}\sum_{j \in n A}X_j, A \in \mathscr{A}$. If for the metric $\rho(A, B) = |A \Delta B|$ the metric entropy with inclusion $N_1(\varepsilon, \mathscr{A}, \rho)$ satisfies $\int^1_0(\varepsilon^{-1} \log N_I(\varepsilon, \mathscr{A}, \rho))^{1/2} d\varepsilon < \infty$, then an appropriately smoothed version of the partial-sum process converges weakly to the Brownian process indexed by $\mathscr{A}$. This improves on previous results of Pyke (1983) and of Bass and Pyke (1984) which require stronger conditions on the moments of $X_j$.

Sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class $\mathscr{F}$ of functions are proved under several kinds of conditions on $\mathscr{F}$. These bounds are used to establish laws of the iterated logarithm for this supremum and to obtain rates of convergence in total variation for empirical processes on the integers.