The stationary isotropic Poisson line process was used to derive upper bounds on mean excess network geodesic length in Aldous and Kendall [Adv. in Appl. Probab. 40 (2008) 1–21]. The current paper presents a study of the geometry and fluctuations of near-geodesics in the network generated by the line process. The notion of a “Poissonian city” is introduced, in which connections between pairs of nodes are made using simple “no-overshoot” paths based on the Poisson line process. Asymptotics for geometric features and random variation in length are computed for such near-geodesic paths; it is shown that they traverse the network with an order of efficiency comparable to that of true network geodesics. Mean characteristics and limiting behavior at the center are computed for a natural network flow. Comparisons are drawn with similar network flows in a city based on a comparable rectilinear grid. A concluding section discusses several open problems.

We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, for example, suits disregarded or only the colors of interest. For these features the number of shuffles drops from (3/2) log₂ n to log₂ n. We derive closed formulae and an asymptotic “rule of thumb” formula which is remarkably accurate.

We develop appropriately generalized notions of indexability for problems of dynamic resource allocation where the resource concerned may be assigned more flexibility than is allowed, for example, in classical multi-armed bandits. Most especially we have in mind the allocation of a divisible resource (manpower, money, equipment) to a collection of objects (projects) requiring it in cases where its over-concentration would usually be far from optimal. The resulting project indices are functions of both a resource level and a state. They have a simple interpretation as a fair charge for increasing the resource available to the project from the specified resource level when in the specified state. We illustrate ideas by reference to two model classes which are of independent interest. In the first, a pool of servers is assigned dynamically to a collection of service teams, each of which mans a service station. We demonstrate indexability under a natural assumption that the service rate delivered is increasing and concave in the team size. The second model class is a generalization of the spinning plates model for the optimal deployment of a divisible investment resource to a collection of reward generating assets. Asset indexability is established under appropriately drawn laws of diminishing returns for resource deployment. For both model classes numerical studies provide evidence that the proposed greedy index heuristic performs strongly.

Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649–667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The key idea was to use suitable linear functionals of the noise process in the numerical scheme which allow a higher approximation order to be obtained. Following this approach, a new simplified version of the scheme in the above named reference is proposed and analyzed in this article. The main advantage of the convergence result given here is the higher convergence order for nonlinear parabolic SPDEs with additive noise, although the used numerical scheme is very simple to simulate and implement.

Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the boundary in a specified manner. The data for such a process are a drift vector θ, a nonsingular d×d covariance matrix Σ, and a d×d reflection matrix R. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions for positive recurrence are easy to formulate for d=2, but not for d>2.

Associated with the pair (θ, R) are fluid paths, which are solutions of deterministic equations corresponding to the random equations of the SRBM. A standard result of Dupuis and Williams [Ann. Probab. 22 (1994) 680–702] states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. Employing this result, El Kharroubi, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229–253, Math. Methods Oper. Res. 56 (2002) 243–258] gave sufficient conditions on (θ, Σ, R) for positive recurrence for d=3; Bramson, Dai and Harrison [Ann. Appl. Probab. 20 (2009) 753–783] showed that these conditions are, in fact, necessary.

Relatively little is known about the recurrence behavior of SRBMs for d>3. This pertains, in particular, to necessary conditions for positive recurrence. Here, we provide a family of examples, in d=6, with θ=(−1, −1, …, −1)^T, Σ=I and appropriate R, that are positive recurrent, but for which a linear fluid path diverges to infinity. These examples show in particular that, for d≥6, the converse of the Dupuis–Williams result does not hold.

We study how the structure of the interaction graph of a game affects the existence of pure Nash equilibria. In particular, for a fixed interaction graph, we are interested in whether there are pure Nash equilibria arising when random utility tables are assigned to the players. We provide conditions for the structure of the graph under which equilibria are likely to exist and complementary conditions which make the existence of equilibria highly unlikely. Our results have immediate implications for many deterministic graphs and generalize known results for random games on the complete graph. In particular, our results imply that the probability that bounded degree graphs have pure Nash equilibria is exponentially small in the size of the graph and yield a simple algorithm that finds small nonexistence certificates for a large family of graphs. Then we show that in any strongly connected graph of n vertices with expansion (1+Ω(1))log₂(n) the distribution of the number of equilibria approaches the Poisson distribution with parameter 1, asymptotically as n→+∞.

In order to obtain a refined characterization of the degree of connectivity associated with the existence of equilibria, we also study the model in the random graph setting. In particular, we look at the case where the interaction graph is drawn from the Erdős–Rényi, G(n, p), model where each edge is present independently with probability p. For this model we establish a double phase transition for the existence of pure Nash equilibria as a function of the average degree pn, consistent with the nonmonotone behavior of the model. We show that when the average degree satisfies np>(2+Ω(1))log_e(n), the number of pure Nash equilibria follows a Poisson distribution with parameter 1, asymptotically as n→∞. When 1/n≪np<(0.5−Ω(1))log_e(n), pure Nash equilibria fail to exist with high probability. Finally, when np=O(1/n) a pure Nash equilibrium exists with constant probability.

This article is concerned with the fluctuations and the concentration properties of a general class of discrete generation and mean field particle interpretations of nonlinear measure valued processes. We combine an original stochastic perturbation analysis with a concentration analysis for triangular arrays of conditionally independent random sequences, which may be of independent interest. Under some additional stability properties of the limiting measure valued processes, uniform concentration properties, with respect to the time parameter, are also derived. The concentration inequalities presented here generalize the classical Hoeffding, Bernstein and Bennett inequalities for independent random sequences to interacting particle systems, yielding very new results for this class of models.

We illustrate these results in the context of McKean–Vlasov-type diffusion models, McKean collision-type models of gases and of a class of Feynman–Kac distribution flows arising in stochastic engineering sciences and in molecular chemistry.

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose.

We also show that in the k-nearest neighbor model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.

We develop a technique that provides a lower bound on the speed of transient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and regeneration time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic [Probab. Theory Related Fields. 122 (2002) 567–592] prove an upper bound of the form b/(b+δ) for the speed on the b-ary tree, where δ is the reinforcement parameter. For δ>1 we provide a lower bound of the form γ²b/(b+δ), where γ is the survival probability of an associated branching process.

It is well known how to determine the price of perpetual American options if the underlying stock price is a time-homogeneous diffusion. In the present paper we consider the inverse problem, that is, given prices of perpetual American options for different strikes, we show how to construct a time-homogeneous stock price model which reproduces the given option prices.

In this paper, we present applications of the calculus developed in Collins and Nechita [Comm. Math. Phys. 297 (2010) 345–370] and obtain an exact formula for the moments of random quantum channels whose input is a pure state thanks to Gaussianization methods. Our main application is an in-depth study of the random matrix model introduced by Hayden and Winter [Comm. Math. Phys. 284 (2008) 263–280] and used recently by Brandao and Horodecki [Open Syst. Inf. Dyn. 17 (2010) 31–52] and Fukuda and King [J. Math. Phys. 51 (2010) 042201] to refine the Hastings counterexample to the additivity conjecture in quantum information theory. This model is exotic from the point of view of random matrix theory as its eigenvalues obey two different scalings simultaneously. We study its asymptotic behavior and obtain an asymptotic expansion for its von Neumann entropy.

We derive the joint density of market weights, at fixed times and suitable stopping times, of the volatility-stabilized market models introduced by Fernholz and Karatzas in [Ann. Finan. 1 (2005) 149–177]. The argument rests on computing the exit density of a collection of independent Bessel-square processes of possibly different dimensions from the unit simplex. We show that the law of the market weights is the same as that of the multi-allele Wright–Fisher diffusion model, well known in population genetics. Thus, as a side result, we furnish a novel proof of the transition density function of the Wright–Fisher model which was originally derived by Griffiths by bi-orthogonal series expansion.