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Register a new userStationary cocycles for the corner growth model
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We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, define solutions to variational formulas that characterize limit shapes, and yield new results for Busemann functions, geodesics and the competition interface.
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We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as boundary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles will be used to prove results about semi-infinite geodesics and the competition interface.
We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. In a previous paper we constructed stationary cocycles and Busemann functions for this model. Using these objects, we prove new results on the competition interface, on existence, uniqueness, and coalescence of directional semi-infinite geodesics, and on nonexistence of doubly infinite geodesics.
In recent years, the application potential of visible light communication (VLC) technology as an alternative and supplement to radio frequency (RF) technology has attracted peoples attention. The study of the underlying VLC channel is the basis for designing the VLC communication system. In this paper, a new non-stationary geometric street corner model is proposed for vehicular VLC (VVLC) multiple-input multiple-output (MIMO) channel. The proposed model takes into account changes in vehicle speed and direction. The category of scatterers includes fixed scatterers and mobile scatterers (MS). Based on the proposed model, we derive the channel impulse response (CIR) and explore the statistical characteristics of the VVLC channel. The channel gain and root mean square (RMS) delay spread of the VVLC channel are studied. In addition, the influence of velocity change on the statistical characteristics of the model is also investigated. The proposed channel model can guide future vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) optical communication system design.
We consider Gaussian approximation in a variant of the classical Johnson-Mehl birth-growth model with random growth speed. Seeds appear randomly in $mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed and a weight function $h:mathbb{R}^d to [0,infty)$, we provide sufficient conditions for a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.
The forecasting problem for a stationary and ergodic binary time series ${X_n}_{n=0}^{infty}$ is to estimate the probability that $X_{n+1}=1$ based on the observations $X_i$, $0le ile n$ without prior knowledge of the distribution of the process ${X_n}$. It is known that this is not possible if one estimates at all values of $n$. We present a simple procedure which will attempt to make such a prediction infinitely often at carefully selected stopping times chosen by the algorithm. We show that the proposed procedure is consistent under certain conditions, and we estimate the growth rate of the stopping times.
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