The abelian sandpile model on randomly rooted graphs and self-similar groups

The Abelian sandpile model is an archetypical model of the physical phenomenon of self-organized criticality. It is also well studied in combinatorics under the name of chip-firing games on graphs. One of the main open problems about this model is to provide rigorous mathematical explication for predictions about the values of its critical exponents, originating in physics. The model was initially defined on the cubic lattices ℤ d , but the only case where the value of some critical exponent has been established so far is the case of the infinite regular tree—the Bethe lattice. This paper is devoted to the study of the abelian sandpile model on a large class of graphs that serve as approximations to Julia sets of postcritically finite polynomials and occur naturally in the study of automorphism group actions on infinite rooted trees. While different from the square lattice, these graphs share many of its geometric properties: they are of polynomial growth, have one end, and random walks on them are recurrent. This ensures that the behaviour of sandpiles on them is quite different from that observed on the infinite tree. We compute the critical exponent for the decay of mass of sand avalanches on these graphs and prove that it is inversely proportional to the rate of polynomial growth of the graph, thus providing the first rigorous derivation of the critical exponent different from the mean-field (the tree) value

On the origin of 140 GHz emission from the 4 July 2012 solar flare

The sub-THz event observed on the 4 July 2012 with the Bauman Moscow State Technical University Radio Telescope RT-7.5 at 93 and 140~GHz as well as Kislovodsk and Mets\"ahovi radio telescopes, Radio Solar Telescope Network (RSTN), GOES, RHESSI, and SDO orbital stations is analyzed. The spectral flux between 93 and 140 GHz has been observed increasing with frequency. On the basis of the SDO/AIA data the differential emission measure has been calculated. It is shown that the thermal coronal plasma with the temperature above 0.5~MK cannot be responsible for the observed sub-THz flare emission. The non-thermal gyrosynchrotron mechanism can be responsible for the microwave emission near $10$~GHz but the observed millimeter spectral characteristics are likely to be produced by the thermal bremsstrahlung emission from plasma with a temperature of about 0.1~MK.Comment: 18 pages, 6 figure

Subset currents on free groups

We introduce and study the space of \emph{subset currents} on the free group $F_N$. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. While ordinary geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$, and, more generally, in a word-hyperbolic group. The concept of a subset current is related to the notion of an "invariant random subgroup" with respect to some conjugacy-invariant probability measure on the space of closed subgroups of a topological group. If we fix a free basis $A$ of $F_N$, a subset current may also be viewed as an $F_N$-invariant measure on a "branching" analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$.Comment: updated version; to appear in Geometriae Dedicat

The ESA "Plasma Laboratory in Space" study

The European Space Agency has initiated, in the context of its General Studies Programme, a study of the possible use of space for studies in pure and applied plasma physics, in areas not traditionally covered by “space plasma physics”. A team of experts has been set-up to review a broad range of area including industrial plasma physics and pure plasma physics, astrophysical and solar-terrestrial areas. A set of experiments have been identified that can potentially provide access to new phenomena and to allow advances in several fields of plasma science. These experiments concern phenomena on spatial scale (102 to104 m) intermediate between what is achievable on ground experiment and usual solar system plasma observations

Absolutely continuous spectrum for multi-type Galton Watson trees

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.Comment: to appear in Annales Henri Poincar\'

Effective-Range Expansion of the Neutron-Deuteron Scattering Studied by a Quark-Model Nonlocal Gaussian Potential

The S-wave effective range parameters of the neutron-deuteron (nd) scattering are derived in the Faddeev formalism, using a nonlocal Gaussian potential based on the quark-model baryon-baryon interaction fss2. The spin-doublet low-energy eigenphase shift is sufficiently attractive to reproduce predictions by the AV18 plus Urbana three-nucleon force, yielding the observed value of the doublet scattering length and the correct differential cross sections below the deuteron breakup threshold. This conclusion is consistent with the previous result for the triton binding energy, which is nearly reproduced by fss2 without reinforcing it with the three-nucleon force.Comment: 21 pages, 6 figures and 6 tables, submitted to Prog. Theor. Phy

Наблюдения Солнца на радиотелескопе РТ-7.5 МГТУ им. Н.Э. Баумана и моделирование миллиметрового излучения солнечной вспышки 2 апреля 2017 года

Presently, in the territory of Russia active millimeter solar observations are under way via Bauman Moscow State Technical University (BMSTU) RT-7.5 radio telescope (Shustikov et al., 2015, Smirnova et al., 2016; Rozanov 1981; Smirnova et al., 2013; Ryzhov et al., 2014). Rozanov 1981; Smirnova et al. 2013; Рыжов и др. 2014). A radio telescope antenna of the 7.75 m diameter with its unique receiving equipment mounted for frequencies of 93 and 140 GHz, (3.2 and 2.2 mm) allows observing the Sun in the mode of tracking area, or scanning, simultaneously at two frequencies. Such observations are a source of the most valuable information on flares - the most powerful solar activity effects. It is impossible to understand a nature of the energy release mechanisms in the solar flares when there is no adequate diagnosis of physical parameters of plasma and magnetic fields. Meanwhile, sub-THz (0.1-1 THz) radio observations make it possible to diagnose the most inaccessible areas for the Sun researchers, i.e. the chromosphere. The ground-based observations in the millimeter range are rare because of high sensitivity requirements for receiving equipment and significant atmosphere disturbances, while the appropriate space projects are just under development. The paper objective is to investigate a nature of the sub-THz emission from the flare with the positive spectral tilt recorded on the RT-7.5 radio telescope on April 2, 2017. To solve this problem, we used observations in the x-ray, ultraviolet, and microwave ranges. Based on the model of the flare Machado et.al. (1980) Chromosphere, a coronal plasma contribution to the millimeter radio emission has been estimated. This approach allows us not only to take a fresh look at the results obtained earlier, but also makes it possible to diagnose in-depth the physical parameters of the flare plasma.В настоящее время на территории России активно проводятся миллиметровые наблюдения Солнца на радиотелескопе РТ-7.5 МГТУ им. Н.Э. Баумана (Шустиков и др. 2015; Смирнова и др. 2016; Rozanov 1981; Smirnova et al. 2013; Рыжов и др. 2014). Антенна радиотелескопа диаметром 7,75 м с установленной на ней уникальной приемной аппаратурой для частот 93 и 140 ГГц, (3.2 и 2.2 мм) позволяет наблюдать Солнце в режиме ведения области, или сканирования, одновременно на двух частотах. Такие наблюдения являются источником ценнейшей информации о вспышках — наиболее мощных проявлениях солнечной активности. Понять природу механизмов энерговыделения в солнечных вспышках без адекватной диагностики физических параметров плазмы и магнитных полей невозможно. Между тем суб-терагерцовые (0.1-1 TГц) радионаблюдения позволяют диагностировать наиболее труднодоступную область для исследователей Солнца - хромосферу. Наземные наблюдения в миллиметровом диапазоне проводятся редко из-за высоких требований к чувствительности приемной аппаратуры и значительных атмосферных помех, в то время как соответствующие космические проекты находятся лишь на стадии разработки. Цель данной работы – исследовать природу суб-терагерцового излучения вспышки с положительным наклоном спектра 02 апреля 2017 г., зарегистрированной на радиотелескопе РТ-7.5. Для решения поставленной задачи использовались наблюдения в рентгеновском, ультрафиолетовом и микроволновом диапазонах. На основе модели вспышечной хромосферы Machado et.al. (1980), оценен вклад корональной плазмы в миллиметровое радиоизлучение. Такой подход позволяет не только по-новому взглянуть на полученные ранее результаты, но и дает возможность провести углубленную диагностику физических параметров вспышечной плазмы

Green Functions On Trees With Finitely Many Cone Types

We prove that the Green function of a nearest neighbour random walk on a tree with finitely many cone types is algebraic. Analysis of this function for a special class of trees associated to groups leads to new estimates of spectral radii of groups with finitely many cone types. Typeset by A M S-T E X 1. Introduction Let # be an infinite, locally finite, connected, simple graph with the vertex set V (#). A random walk on # is given by a stochastic transition matrix P = p(x, y) x,y#V (#) . In this paper we will only consider nearest neighbour random walks, i.e., such that p(x, y) #= 0 implies x # y. We also suppose P irreducible, i.e., for every x, y # V (#) there exists n # N such that the probability to go from x to y in n steps p (n) (x, y) is positive. The transition matrix P acts on real valued functions on V (#) by P#(x) = X y#V (#) p(x, y)#(y) . A function # is called harmonic if P# = #. The Green kernel on # associated with the random walk P is defined ..