Brownian coagulation and a version of Smoluchowski's equation on the circle

We introduce a one-dimensional stochastic system where particles perform independent diffusions and interact through pairwise coagulation events, which occur at a nontrivial rate upon collision. Under appropriate conditions on the diffusion coefficients, the coagulation rates and the initial distribution of particles, we derive a spatially inhomogeneous version of the mass flow equation as the particle number tends to infinity. The mass flow equation is in one-to-one correspondence with Smoluchowski's coagulation equation. We prove uniqueness for this equation in a broad class of solutions, to which the weak limit of the stochastic system is shown to belong.Comment: Published in at http://dx.doi.org/10.1214/09-AAP633 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Metastability in a condensing zero-range process in the thermodynamic limit

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density in [J. Beltran, C. Landim, Probab. Theory Related Fields, 152(3-4):781-807, 2012], and we follow the martingale approach developed there and in subsequent publications. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.Comment: 55 pages, 1 figur

Phase transition for the dilute clock model

We prove that phase transition occurs in the dilute ferromagnetic nearest-neighbour $q$-state clock model in $\mathbb{Z}^d$, for every $q\geq 2$ and $d\geq 2$. This follows from the fact that the Edwards-Sokal random-cluster representation of the clock model stochastically dominates a supercritical Bernoulli bond percolation probability, a technique that has been applied to show phase transition for the low-temperature Potts model. The domination involves a combinatorial lemma which is one of the main points of this article.Comment: 14 pages, 2 figure

Secondary production of <i>Chasmagnathus granulatus</i> (Crustacea; Decapoda) in a Ramsar Site from Argentina

Secondary production of Chasmagnathus granulatus was calculated at the Refugio de Vida Silvestre Bahía Samborombón, Argentina (36° 16′ S and 57° 06′ W). Sampling was conducted on nine occasions between March 2001 and February 2003, crabs were collected by hand, physico-chemical variables, granulometry and organic matter contents of the sediments were registered. Crabs were classified as male, female and undifferentiated, measured (total carapace width: CW) and weighed (wet and dry weight: DW at 60 °C, during 48 hours). A correlation analysis between CW and DW was made. Morphometric growth of C. granulatus was by the application of the power function (y = a x b), where the carapace width (CW) was used as an independent variable. Males, females and undifferentiated individuals were analysed separately as well as all together as a group. The data were fitted indicating a positive allometry (constant of allometry b > 3), the males showing the greatest allometric value. The individuals (n = 957 juveniles and adults) were separated in cohorts by the polymodal width-frequency distribution converted into normal curves. Three cohorts were found during the whole study period, and two cohorts coexisting in each sampling date. Ovigerous females were caught on December 2001, 2002 and February 2003. The size-frequency method was used to estimate the annual production. The major contribution to production was carried out by the mature individuals, in particular those with size between 25 and 30 mm, but on the other hand, only few individuals measuring from 10 to 20 mm were collected. The annual production of C. granulatus was estimated in 7.76 g.m-2. The biomass (expressed as total dry weight) varied between 0.55 and 1.85 g.m-2, with the greater values being registered during autumn and spring, and the lower values during summer.Foram calculadas as primeiras estimativas da produ&ccedil;&atilde;o secund&aacute;ria de Chasmagnathus granulatus no Ref&uacute;gio da Vida Silvestre Bah&iacute;a Samboromb&oacute;n, Argentina (36&deg; 16&rsquo; S e 57&deg; 06&rsquo; W). As amostragens foram realizadas em nove ocasi- &otilde;es, entre o per&iacute;odo de mar&ccedil;o de 2001 a fevereiro de 2003. Os caranguejos foram coletados manualmente. Vari&aacute;veis f&iacute;sico-qu&iacute;micas, granulometria e teor de mat&eacute;ria org&acirc;nica dos sedimentos foram analisados. O caranguejos (n = 957, jovens e adultos) foram classificados em: machos, f&ecirc;meas e indiferenciados; foram mensurados (Largura da Carapa&ccedil;a, LC) e pesados (peso &uacute;mido e seco, PS a 60 &deg;C, 48 horas). Uma an&aacute;lise de correla&ccedil;&atilde;o entre LC vs. PS foi efetuada. O crescimento morfom&eacute;trico de C. granulatus foi analisado pela aplica&ccedil;&atilde;o da fun&ccedil;&atilde;o potencial (PS = a LC b ), sendo que machos, f&ecirc;meas e indiv&iacute;duos com sexo n&atilde;o diferenciado foram analisados separadamente e em conjunto. O ajuste dos dados indicou uma alometria positiva, com os machos apresentando o maior valor. Os indiv&iacute;duos foram separados em coortes por meio da convers&atilde;o de curvas de normalidade de uma distribui&ccedil;&atilde;o de freq&uuml;&ecirc;ncia polimodal. Tr&ecirc;s coortes foram encontradas durante todo o per&iacute;odo de estudo, e duas coexistindo em cada data de amostragem. F&ecirc;meas ov&iacute;- geras foram coletadas em dezembro de 2001 e 2002 e em fevereiro de 2003. Utilizou-se o m&eacute;todo da distribui&ccedil;&atilde;o de freq&uuml;&ecirc;ncia de tamanho para estimar a produ&ccedil;&atilde;o anual. A maior contribui&ccedil;&atilde;o para a produ&ccedil;&atilde;o &eacute; dada pelos indiv&iacute;duos sexualmente maduros, particularmente por aqueles com tamanhos entre 25 e 30 mm. A produ&ccedil;&atilde;o anual foi estimada em 7,76 g.m-2. A biomassa, expressa em peso seco, variou entre 0,55 e 1,85 g.m-2, com maiores valores registrados no outono e na primavera, e menores valores registrados no ver&atilde;o

Conditional Distribution of Heavy Tailed Random Variables on Large Deviations of their Sum

It is known that large deviations of sums of subexponential random variables are most likely realised by deviations of a single random variable. In this article we give a detailed picture of how subexponential random variables are distributed when a large deviation of their sum is observed.Comment: Results on local conditioning adde

Zero-range processes with rapidly growing rates

We provide two methods to construct zero-range processes with superlinear rates on ${\mathbb Z}^d$. In the first method these rates can grow very fast, if either the dynamics and the initial distribution are translation invariant or if only nearest neigbour translation invariant jumps are permitted, in the one-dimensional lattice. In the second method the rates cannot grow as fast but more general dynamics are allowed.Comment: 33 page

Fecundity of <i>Uca uruguayensis</i> and <i>Chasmagnathus granulatus</i> (Decapoda, Brachyura) from the "Refugio de Vida Silvestre" Bahía Samborombón, Argentina

The aim of the present work conducted at the Refugio de Vida Silvestre Bahía Samborombón is to analyse the most relevant aspects of the fecundity of Chasmagnathus granulatus and Uca uruguayensis. Samplings were carried out from March 2001 to February 2003. Ovigerous females of U. uruguayensis (N = 13) and C. granulatus (N = 25) were found during spring and summer, their sizes (CW) varied from 9.1 to 11.7 mm for the former species and from 22.8 to 32.4 mm for the latter. The egg diameter in U. uruguayensis ranged from 245 to 260 ?m for embryos in the early stage of development and from 250 to 345 ?m for those in mid-developmental stage, while in C. granulatus from 250 to 345 μm and from 260 to 365 ?m respectively. Fecundity varied from 1126 to 6745 eggs/brood in U. uruguayensis and 15688-57418 eggs/brood in C. granulatus. For those females with broods in mid-developmental stage, several relationships were made. For U. uruguayensis the best correlation coefficients were obtained for the relationships: female weight vs. egg mass weight and carapace width vs. egg mass weight; for C. granulatus the best association was obtained between female size and the egg number and the egg mass weight.O objetivo do presente trabalho foi analisar os aspectos mais relevantes da fecundidade de Chasmagnathus granulatus e Uca uruguayensis no Ref&uacute;gio da Vida Silvestre Bahia Samboromb&oacute;n. As amostragens foram realizadas de mar&ccedil;o de 2001 a fevereiro de 2003. As f&ecirc;meas ov&iacute;geras de U. uruguayensis (N = 13) e de C. granulatus (N = 25) foram capturadas na primavera e ver&atilde;o. A largura da carapa&ccedil;a (LC) de U. uruguayensis variou de 9.1 a 11.7 mm, e de 22.8 a 32.4 mm para C. granulatus. O di&acirc;metro dos ovos de U. uruguayensis variou de 245 a 260 &micro;m para embri&otilde;es em est&aacute;gio de desenvolvimento inicial e de 250 a 345 &micro;m para aqueles em est&aacute;gio intermedi&aacute;rio; para C. granulatus as varia&ccedil;&otilde;es foram de 250 a 345 &micro;m e de 260 a 365 &micro;m, respectivamente. A fecundidade de U. uruguayensis variou de 1126 a 6745 ovos/desova e para C. granulatus de 15688 a 57418 ovos/desova. Para as f&ecirc;meas com massa de ovos em est&aacute;gio de desenvolvimento intermedi&aacute;rio foram efetuadas v&aacute;rias rela&ccedil;&otilde;es: para U. uruguayensis os melhores coeficientes de correla&ccedil;&atilde;o foram obtidos nas rela&ccedil;&otilde;es: peso da f&ecirc;mea vs. peso da massa de ovos, e largura da carapa&ccedil;a vs. peso da massa de ovos. Para C. granulatus, a melhor associa&ccedil;&atilde;o foi obtida entre o n&uacute;mero de ovos e o peso da massa de ovos

Gaussian random permutation and the boson point process

We construct an infinite volume spatial random permutation (χ,σ), where χ⊂ℝd is a point process and σ:χ→χ is a permutation (bijection), associated to the formal Hamiltonian H(χ,σ)=∑_x∈χ‖x−σ(x)‖2. The measures are parametrized by the density ρ of points and the temperature α. Feynman (1953) related spatial random permutations with boson systems and proposed that Bose-Einstein condensation occurs precisely when infinite cycles appear in the corresponding random permutation. Each finite cycle of σ induces a loop of points of χ. For ρ ≤ ρc we define (χ, σ) as a Poisson process of finite unrooted loops that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (2004). We also construct Gaussian random interlacements, a Poisson process of double-infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (2007). For d ≥ 3 and ρ > ρc we define (χ, σ) as the superposition of independent realizations of the Gaussian loop soup at density ρc and the Gaussian random interlacements at density ρ − ρc and call it a Gaussian random permutation at density ρ and temperature α. The resulting measure is Gibbs for the Hamiltonian H and the point marginal χ has the same distribution as the boson point process introduced by Macchi (1975) in the subcritical case and by Tamura-Ito (2007) in the supercritical case. Bose-Einstein condensation occurs when the Gaussian random permutation exhibits infinite trajectories.Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Yuhjtman, Sergio Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Gibbs measures over permutations of point processes with low density

We study a model of spatial random permutations over a discrete set of points. Formally, a permutation σ is sampled proportionally to the weight exp{−α∑_xV(σ(x)−x)}, where α>0 is the temperature and V is a non-negative and continuous potential. The most relevant case for physics is when V(x)=‖x‖^2, since it is related to Bose-Einstein condensation through a representation introduced by Feynman in 1953. In the context of statistical mechanics, the weights (1) define a probability when the set of points is finite, but the construction associated to an infinite set is not trivial and may fail without appropriate hypotheses. The first problem is to establish conditions for the existence of such a measure at infinite volume when the set of points is infinite. Once existence is derived, we are interested in establishing its uniqueness and the cycle structure of a typical permutation. We here consider the large temperature regime when the set of points is a Poisson point process in ℤ^d with intensity ρ∈(0,1/2), and the potential verifies some regularity conditions. In particular, we prove that if α is large enough, for almost every realization of the point process, there exists a unique Gibbs measure that concentrates on finite cycle permutations. We then extend these results to the continuous setting, when the set of points is given by a Poisson point process in ℝ^d with low enough intensity.Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Frevenza Maestrone, Nicolas Federico. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin