Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.Comment: 18 pages, 4 figures. Some sections rewritten (including title and abstract), numerical results added, references added. Accepted for publication in J. Stat. Phy

Simple and Effective Type Check Removal through Lazy Basic Block Versioning

Dynamically typed programming languages such as JavaScript and Python defer type checking to run time. In order to maximize performance, dynamic language VM implementations must attempt to eliminate redundant dynamic type checks. However, type inference analyses are often costly and involve tradeoffs between compilation time and resulting precision. This has lead to the creation of increasingly complex multi-tiered VM architectures. This paper introduces lazy basic block versioning, a simple JIT compilation technique which effectively removes redundant type checks from critical code paths. This novel approach lazily generates type-specialized versions of basic blocks on-the-fly while propagating context-dependent type information. This does not require the use of costly program analyses, is not restricted by the precision limitations of traditional type analyses and avoids the implementation complexity of speculative optimization techniques. We have implemented intraprocedural lazy basic block versioning in a JavaScript JIT compiler. This approach is compared with a classical flow-based type analysis. Lazy basic block versioning performs as well or better on all benchmarks. On average, 71% of type tests are eliminated, yielding speedups of up to 50%. We also show that our implementation generates more efficient machine code than TraceMonkey, a tracing JIT compiler for JavaScript, on several benchmarks. The combination of implementation simplicity, low algorithmic complexity and good run time performance makes basic block versioning attractive for baseline JIT compilers

Time-causal and time-recursive spatio-temporal receptive fields

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, based on a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision. We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain much faster temporal response properties (shorter temporal delays) compared to a uniform distribution. Specifically, these kernels converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales, thereby allowing for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter. We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.Comment: 39 pages, 12 figures, 5 tables in Journal of Mathematical Imaging and Vision, published online Dec 201

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172 pages, low quality figures, better quality figures upon request to the authors, comments welcom