## Recursos de colección

1. #### The largest component in a subcritical random graph with a power law degree distribution

Janson, Svante
It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent γ>3, the largest component is of order n1/(γ−1). More precisely, the order of the largest component is approximatively given by a simple constant times the largest vertex degree. These results are extended to several other random graph models with power law degree distributions. This proves a conjecture by Durrett.

2. #### On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

Pittel, B. G.
A uniformly random graph on n vertices with a fixed degree sequence, obeying a γ subpower law, is studied. It is shown that, for γ>3, in a subcritical phase with high probability the largest component size does not exceed n1/γ+ɛn, ɛn=O(ln ln n/ln n), 1/γ being the best power for this random graph. This is similar to the best possible n1/(γ−1) bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.

3. #### Central limit theorem for branching random walks in random environment

Yoshida, Nobuo
We consider branching random walks in d-dimensional integer lattice with time–space i.i.d. offspring distributions. When d≥3 and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.

4. #### Stein’s method for discrete Gibbs measures

Eichelsbacher, Peter; Reinert, Gesine
Stein’s method provides a way of bounding the distance of a probability distribution to a target distribution μ. Here we develop Stein’s method for the class of discrete Gibbs measures with a density eV, where V is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) 30–42]. We obtain estimates of the approximation to a grand-canonical Gibbs ensemble. As side results, we slightly improve on the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds for Poisson approximation to the sum...

5. #### Disordered pinning models and copolymers: Beyond annealed bounds

Toninelli, Fabio Lucio
We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9–13], pinning and wetting models in various dimensions, and the Poland–Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, for example, if the disorder distribution is Gaussian. In particular, for pinning...

6. #### Central limit theorem for a many-server queue with random service rates

Atar, Rami
Given a random variable N with values in ℕ, and N i.i.d. positive random variables {μk}, we consider a queue with renewal arrivals and N exponential servers, where server k serves at rate μk, under two work conserving routing schemes. In the first, the service rates {μk} need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that...

7. #### Bayesian nonparametric estimators derived from conditional Gibbs structures

Lijoi, Antonio; Prünster, Igor; Walker, Stephen G.
We consider discrete nonparametric priors which induce Gibbs-type exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required.

8. #### Computable exponential bounds for screened estimation and simulation

Kontoyiannis, Ioannis; Meyn, Sean P.
Suppose the expectation E(F(X)) is to be estimated by the empirical averages of the values of F on independent and identically distributed samples {Xi}. A sampling rule called the “screened” estimator is introduced, and its performance is studied. When the mean E(U(X)) of a different function U is known, the estimates are “screened,” in that we only consider those which correspond to times when the empirical average of the {U(Xi)} is sufficiently close to its known mean. As long as U dominates F appropriately, the screened estimates admit exponential error bounds, even when F(X) is heavy-tailed. The main results are...

9. #### On the disconnection of a discrete cylinder by a biased random walk

Windisch, David
We consider a random walk on the discrete cylinder (ℤ/Nℤ)d×ℤ, d≥3 with drift N−dα in the ℤ-direction and investigate the large N-behavior of the disconnection time TNdisc, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent α is strictly greater than 1, the asymptotic behavior of TNdisc remains N2d+o(1), as in the unbiased case considered by Dembo and Sznitman, whereas for α<1, the asymptotic behavior of TNdisc becomes exponential in N.

10. #### Analytic crossing probabilities for certain barriers by Brownian motion

Kahale, Nabil
We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval [0, T] via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class of barriers on [0, T] by proving a Schwartz distribution version of the method of images. Analytic expressions for crossing probabilities and related densities are given for new explicit and semi-explicit barriers.

11. #### Large deviations and a Kramers’ type law for self-stabilizing diffusions

Herrmann, Samuel; Imkeller, Peter; Peithmann, Dierk
We investigate exit times from domains of attraction for the motion of a self-stabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is the effect of including an ensemble-average attraction in addition to the usual state-dependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers’ type law for the particle’s exit from the potential’s domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that the exit law for the self-stabilizing diffusion coincides with the...

12. #### Efficient rare-event simulation for the maximum of heavy-tailed random walks

Blanchet, Jose; Glynn, Peter
Let (Xn : n≥0) be a sequence of i.i.d. r.v.’s with negative mean. Set S0=0 and define Sn=X1+⋯+Xn. We propose an importance sampling algorithm to estimate the tail of M=max {Sn : n≥0} that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund’s (1979) algorithm....

13. #### Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model

Coupier, David
A d-dimensional ferromagnetic Ising model on a lattice torus is considered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein–Chen method. The rate of the Poisson approximation and the speed of convergence to it are defined and make sense for the model. Thus, the two sufficient conditions are traduced in terms of the magnetic field and the pair potential. In particular, the Poisson approximation holds even if both potentials diverge.

14. #### The emergence of the deterministic Hodgkin–Huxley equations as a limit from the underlying stochastic ion-channel mechanism

Austin, Tim D.
In this paper we consider the classical differential equations of Hodgkin and Huxley and a natural refinement of them to include a layer of stochastic behavior, modeled by a large number of finite-state-space Markov processes coupled to a simple modification of the original Hodgkin–Huxley PDE. We first prove existence, uniqueness and some regularity for the stochastic process, and then show that in a suitable limit as the number of stochastic components of the stochastic model increases and their individual contributions decrease, the process that they determine converges to the trajectory predicted by the deterministic PDE, uniformly up to finite time...

15. #### Perfect simulation for a class of positive recurrent Markov chains

Connor, Stephen B.; Kendall, Wilfrid S.

16. #### Survival and complete convergence for a spatial branching system with local regulation

Birkner, Matthias; Depperschmidt, Andrej
We study a discrete time spatial branching system on ℤd with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the nontrivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model. Along the way we classify the equilibria and their domain of attraction for the corresponding deterministic coupled map lattice on ℤd.

17. #### HJB equations for certain singularly controlled diffusions

Atar, Rami; Budhiraja, Amarjit; Williams, Ruth J.
Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}}^{d}$ with nonempty interior, we consider a control problem in which the state process W and the control process U satisfy ¶ $W_{t}=w_{0}+\int_{0}^{t}\vartheta(W_{s})\,ds+\int_{0}^{t}\sigma(W_{s})\,dZ_{s}+GU_{t}\in \mathcal{W},\qquad t\ge0,$ ¶ where Z is a standard, multi-dimensional Brownian motion, $\vartheta,\sigma\in C^{0,1}(\mathcal{W})$ , G is a fixed matrix, and $w_{0}\in\mathcal{W}$ . The process U is locally of bounded variation and has increments in a given closed convex cone $\mathcal{U}\subset{\mathbb{R}}^{p}$ . Given $g\in C(\mathcal{W})$ , κ∈ℝp, and α>0, consider the objective that is to minimize the cost ¶ $J(w_{0},U)\doteq\mathbb{E}\biggl[\int_{0}^{\infty}e^{-\alpha s}g(W_{s})\,ds+\int_{[0,\infty)}e^{-\alpha s}\,d(\kappa\cdot U_{s})\biggr]$ ¶ over the admissible controls U. Both g and κ⋅u ( $u\in\mathcal{U}$ ) may take positive...

18. #### Isoperimetric inequalities and mixing time for a random walk on a random point process

Caputo, Pietro; Faggionato, Alessandra
We consider the random walk on a simple point process on ℝd, d≥2, whose jump rates decay exponentially in the α-power of jump length. The case α=1 corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for α∈(0, d), that the random walk confined to a cubic box of side L has a.s. Cheeger constant of order at least L−1 and mixing time of order L2. For the Poisson point process, we prove that at α=d, there is a transition from diffusive to subdiffusive...

19. #### Analysis of SPDEs arising in path sampling part II: The nonlinear case

Hairer, M.; Stuart, A. M.; Voss, J.
In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite-dimensional analogue of the Langevin equation used in finite-dimensional sampling. In this paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green’s operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolution equations are important for the development of...

20. #### The two-type Richardson model with unbounded initial configurations

Deijfen, Maria; Häggström, Olle
The two-type Richardson model describes the growth of two competing infections on ℤd and the main question is whether both infection types can simultaneously grow to occupy infinite parts of ℤd. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points x=(x1, …, xd) in the hyperplane $\mathcal{H}=\{x\in\mathbb{Z}^{d}:x_{1}=0\}$ is considered. It is shown that, starting from a configuration where all points in $\mathcal{H}\backslash\{\mathbf{0}\}$ are type 1 infected and the origin 0 is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only...

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