Loop measures without transition probabilities

We construct Markov loop measures without assuming the existence of densities for transition probabilities

Permanental fields, loop soups and continuous additive functionals

A permanental field, $\psi=\{\psi(\nu),\nu\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $\psi$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates $\psi$ to continuous additive functionals of $X$ (continuous in $t$), $L=\{L_t^{\nu},(\nu ,t)\in {\mathcal{V}}\times R_+\}$. Sufficient conditions are obtained for the continuity of $L$ on ${\mathcal{V}}\times R_+$. The metric on ${\mathcal{V}}$ is given by a proper norm.Comment: Published in at http://dx.doi.org/10.1214/13-AOP893 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has been added and several inaccuracies have been corrected. To appear in Journal of Theoretical Probabilit

Towards a fully automated computation of RG-functions for the 3-dd O(N) vector model: Parametrizing amplitudes

Within the framework of field-theoretical description of second-order phase transitions via the 3-dimensional O(N) vector model, accurate predictions for critical exponents can be obtained from (resummation of) the perturbative series of Renormalization-Group functions, which are in turn derived --following Parisi's approach-- from the expansions of appropriate field correlators evaluated at zero external momenta. Such a technique was fully exploited 30 years ago in two seminal works of Baker, Nickel, Green and Meiron, which lead to the knowledge of the $\beta$-function up to the 6-loop level; they succeeded in obtaining a precise numerical evaluation of all needed Feynman amplitudes in momentum space by lowering the dimensionalities of each integration with a cleverly arranged set of computational simplifications. In fact, extending this computation is not straightforward, due both to the factorial proliferation of relevant diagrams and the increasing dimensionality of their associated integrals; in any case, this task can be reasonably carried on only in the framework of an automated environment. On the road towards the creation of such an environment, we here show how a strategy closely inspired by that of Nickel and coworkers can be stated in algorithmic form, and successfully implemented on the computer. As an application, we plot the minimized distributions of residual integrations for the sets of diagrams needed to obtain RG-functions to the full 7-loop level; they represent a good evaluation of the computational effort which will be required to improve the currently available estimates of critical exponents.Comment: 54 pages, 17 figures and 4 table

Coulomb Gauge QCD, Confinement, and the Constituent Representation

Quark confinement and the genesis of the constituent quark model are examined in nonperturbative QCD in Coulomb gauge. We employ a self-consistent method to construct a quasiparticle basis and to determine the quasiparticle interaction. The results agree remarkably well with lattice computations. They also illustrate the mechanism by which confinement and constituent quarks emerge, provide support for the Gribov-Zwanziger confinement scenario, clarify several perplexing issues in the constituent quark model, and permit the construction of an improved model of low energy QCD.Comment: 43 pages, 14 figures, revtex, uses psfig.st

Dome growth, collapse, and valley fill at Soufrière Hills Volcano, Montserrat, from 1995 to 2013: Contributions from satellite radar measurements of topographic change

Frequent high-resolution measurements of topography at active volcanoes can provide important information for assessing the distribution and rate of emplacement of volcanic deposits and their influence on hazard. At dome-building volcanoes, monitoring techniques such as LiDAR and photogrammetry often provide a limited view of the area affected by the eruption. Here, we show the ability of satellite radar observations to image the lava dome and pyroclastic density current deposits that resulted from 15 years of eruptive activity at Soufrière Hills Volcano, Montserrat, from 1995 to 2010. We present the first geodetic measurements of the complete subaerial deposition field on Montserrat, including the lava dome. Synthetic aperture radar observations from the Advanced Land Observation Satellite (ALOS) and TanDEM-X mission are used to map the distribution and magnitude of elevation changes. We estimate a net dense-rock equivalent volume increase of 108 ± 15M m3 of the lava dome and 300 ± 220M m3 of talus and subaerial pyroclastic density current deposits. We also show variations in deposit distribution during different phases of the eruption, with greatest on-land deposition to the south and west, from 1995 to 2005, and the thickest deposits to the west and north after 2005. We conclude by assessing the potential of using radar-derived topographic measurements as a tool for monitoring and hazard assessment during eruptions at dome-building volcanoes