We consider the discrete Gaussian Free Field in a square box in ( Z ^2 ) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as ( N ). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever ( r_N ) and ( r_N N 0 ), to a Poisson point process with intensity measure ( Z ( dx) e ^ - h d h ), where ( := 2 g ) with g: = 2 π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.
The two-dimensional DGFF is particularly interesting because its continuum version (CGFF) is invariant under conformal transformations of the underlying domain. This offers a framework for analyzing scaling limits of certain critical models. For instance, the level sets of the DGFF on the triangular lattice can be linked with the Schramm-Loewner process SLE @math (Schramm and Sheffield @cite_13 ), the height function associated with domino tilings scales to the CGFF (Kenyon @cite_21 ) etc. A complication associated with the CGFF is that, by its scale-invariant nature, the field'' exists only as a random distribution on an appropriate function space.
@cite_21 @cite_13
1512015702 1997019093 2115976335 2006973305
We address an open question, regarding whether a lattice code with lattice decoding (as opposed to maximum-likelihood (ML) decoding) can achieve the additive white Gaussian noise (AWGN) channel capacity. We first demonstrate how minimum mean-square error (MMSE) scaling along with dithering (lattice randomization) techniques can transform the power-constrained AWGN channel into a modulo-lattice additive noise channel, whose effective noise is reduced by a factor of spl radic (1+SNR SNR). For the resulting channel, a uniform input maximizes mutual information, which in the limit of large lattice dimension becomes 1 2 log (1+SNR), i.e., the full capacity of the original power constrained AWGN channel. We then show that capacity may also be achieved using nested lattice codes, the coarse lattice serving for shaping via the modulo-lattice transformation, the fine lattice for channel coding. We show that such pairs exist for any desired nesting ratio, i.e., for any signal-to-noise ratio (SNR). Furthermore, for the modulo-lattice additive noise channel lattice decoding is optimal. Finally, we show that the error exponent of the proposed scheme is lower bounded by the Poltyrev exponent. In a companion paper (see Self-Similarity: Part I-Splines and Operators), we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order gamma. We used these operators to specify regularization functionals for a series of Tikhonov-like least-squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)-based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically self-similar; that is, fractional Brownian motion (fBm) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of fBm, we adopt a distributional formulation due to Gel'fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) p(BH(t)|Y), where Y= BH(k)+n[k] kisinZ are the noisy samples of the fBm BH(t) with Hurst exponent H. We find that the conditional mean is a fractional spline of degree 2H, which proves that this class of functions is indeed optimal for the estimation of fractal-like processes. The result also yields the optimal [minimum mean-square error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering We flnd scaling limits for the sizes of the largest components at criticality for the rank-1 inho- mogeneous random graphs with power-law degrees with exponent ?. We investigate the case where ? 2 (3;4), so that the degrees have flnite variance but inflnite third moment. The sizes of the largest clusters, rescaled by n i(?i2)=(?i1) , converge to hitting times of a 'thinned' Levy process. This process is intimately connected to the general multiplicative coalescents studied in (1) and (2). In particular, we use the results in (2) to show that, when interpreting the location ‚ inside the critical window as time, the limiting process is a multiplicative process with difiusion constant 0 and the entrance boundary describing the size of relative components in the ‚ ! i1 regime proportional to i i i1=(?i1) ¢ i‚1 . A crucial ingredient is the identiflcation of the scaling of the largest connected components in the barely subcritical regime. Our results should be contrasted to the case where the degree exponent ? satisfles ? > 4, so that the third moment is flnite. There, instead, we see that the sizes of the components rescaled by n i2=3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in (1) for the Erd os- Renyi random graph and extended to the present setting in (3, 4). The limit again is a multiplicative coalescent, the only difierence with the limit for ? 2 (3;4) being the initial state, corresponding to the barely subcritical regime. Precise calculation of molecular electronic wavefunctions by methods such as coupled-cluster requires the computation of tensor contractions, the cost of which has polynomial computational scaling with respect to the system and basis set sizes. Each contraction may be executed via matrix multiplication on a properly ordered and structured tensor. However, data transpositions are often needed to reorder the tensors for each contraction. Writing and optimizing distributed-memory kernels for each transposition and contraction is tedious since the number of contractions scales combinatorially with the number of tensor indices. We present a distributed-memory numerical library (Cyclops Tensor Framework (CTF)) that automatically manages tensor blocking and redistribution to perform any user-specified contractions. CTF serves as the distributed-memory contraction engine in Aquarius, a new program designed for high-accuracy and massively-parallel quantum chemical computations. Aquarius implements a range of coupled-cluster and related methods such as CCSD and CCSDT by writing the equations on top of a C++ templated domain-specific language. This DSL calls CTF directly to manage the data and perform the contractions. Our CCSD and CCSDT implementations achieve high parallel scalability on the BlueGene Q and Cray XC30 supercomputer architectures showing that accurate electronic structure calculations can be effectively carried out on top of general distributed-memory tensor primitives. We introduce Cyclops Tensor Framework (CTF), a distributed-memory library for tensor contractions.CTF is able to perform tensor decomposition, redistribution, and contraction at runtime.CTF enables the expression of massively-parallel coupled-cluster methods via a concise tensor contraction interface.The quantum chemistry software suite Aquarius employs CTF to execute two coupled-cluster methods: CCSD and CCSDT.The Aquarius CCSD and CCSDT codes scale well on BlueGene Q and Cray XC30, comparing favorably to NWChem.