For an n-element subset U of $\mathbb {Z}^2$, select x from U according to harmonic measure from infinity, remove x from U and start a random walk from x. If the walk leaves from y when it first enters the rest of U, add y to it. Iterating this procedure constitutes the process we call harmonic activation and transport (HAT).

HAT exhibits a phenomenon we refer to as collapse: Informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion.

To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among n-element subsets of $\mathbb {Z}^2$, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, d? Concerning the former, examples abound for which the harmonic measure is exponentially small in n. We prove that it can be no smaller than exponential in $n \log n$. Regarding the latter, the escape probability is at most the reciprocal of $\log d$, up to a constant factor. We prove it is always at least this much, up to an n-dependent factor.

We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.

In this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.

We present DASHMM, a general library implementing multipole methods (including both Barnes-Hut and the Fast Multipole Method). DASHMM relies on dynamic adaptive runtime techniques provided by the HPX-5 system to parallelize the resulting multipole moment computation. The result is a library that is easy-to-use, extensible, scalable, efficient, and portable. We present both the abstractions defined by DASHMM as well as the specific features of HPX-5 that allow the library to execute scalably and efficiently.

We present RECFMM, a program representation and implementation of a recursive scheme for parallelizing the adaptive fast multipole method (FMM) on shared-memory computers. It achieves remarkable high performance while maintaining mathematical clarity and flexibility. The parallelization scheme signifies the recursion feature that is intrinsic to the FMM but was not well exploited. The program modules of RECFMM constitute a map between numerical computation components and advanced architecture mechanisms. The mathematical structure is preserved and exploited, not obscured nor compromised, by parallel rendition of the recursion scheme. Modern software system—CILK in particular, which provides graph-theoretic optimal scheduling in adaptation to the dynamics in parallel execution—is employed. RECFMM supports multiple algorithm variants that mark the major advances with low-frequency interaction kernels, and includes the asymmetrical version where the source particle ensemble is not necessarily the same as the target particle ensemble. We demonstrate parallel performance with Coulomb and screened Coulomb interactions.

In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in ${{\mathbb{R}}^{d}}$ with the potential ${{\left| x\,-\,y \right|}^{-s}},\,s\,\ge \,1$, where $\left| \cdot \right|$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for $s\,\ge \,2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N,\,N\to \,\infty $, where $L$ is the length of the curve.

We consider the $s$-energy $E({{Z}_{n}};\,s)={{\Sigma }_{i\ne j}}K(\parallel {{z}_{i,n}}\,-\,{{z}_{j,n}}\parallel \,;\,s)$ for point sets $Zn\,=\,\{{{z}_{k,n}}\,:\,k\,=\,0,\,\ldots \,,\,n\} $ on certain compact sets $\Gamma $ in ${{\mathbb{R}}^{d}}$ having finite one-dimensional Hausdorff measure,where

$$K(t;\,s)\,=\,\left\{ _{-\ln \,t,\,\,\,\text{if}\,s\,=\,0,\,}^{{{t}^{-s}},\,\,\,\,\,\,\,\text{if}\,s\,>\,0,} \right\}$$

is the Riesz kernel. Asymptotics for the minimum $s$-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for $s\,\ge \,1$, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as $n\,\to \,\infty $.

One of the questions concerning the Hexagonal Packing Lemma ([1], [3], [4]) is the rate of convergence of Sn. It was suggested in [3] and [4] that Sn = 0(1/n). In the following we prove this conjecture under the additional condition of some "nice" behaviour of the "circle function".