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On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators  [PDF]
Folkmar Bornemann
Mathematics , 2011,
Abstract: By applying an idea of Borodin and Olshanski (2007), we study various scaling limits of determinantal point processes with trace class projection kernels given by spectral projections of selfadjoint Sturm-Liouville operators. Instead of studying the convergence of the kernels as functions, the method directly addresses the strong convergence of the induced integral operators. We show that, for this notion of convergence, the Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and hard-edge scaling limits. This result allows us to give a short and unified derivation of the known formulae for the scaling limits of the classical unitary random matrix ensembles (GUE, LUE/Wishart, JUE/MANOVA).
One-Parametric Family of the Double-Scaling Limits in the Hermitian Matrix Model $Φ^6:$ Onset of Nondissipative Shock Waves  [PDF]
Vadim R. Kudashev,Bulat I. Suleimanov
Physics , 1998,
Abstract: We construct a one-parametric family of the double-scaling limits in the hermitian matrix model $\Phi^6$ for 2D quantum gravity. The known limit of Bresin, Marinari and Parisi belongs to this family. The family is represented by the Gurevich-Pitaevskii solution of the Korteveg-de Vries equation which describes the onset of nondissipative shock waves in media with small dispersion. Numerical simulation of the universal Gurevich-Pitaevskii solution is made.
Universal scaling of Hall resistivity in clean and moderately clean limits for Hg- and Tl-based superconductors  [PDF]
W. N. Kang,Wan-Seon Kim,S. J. Oh,Sung-Ik Lee,D. H. Kim,C. H. Choi,H. -C. Ri,C. W. Chu
Physics , 1999,
Abstract: The mixed-state Hall resistivity and the longitudinal resistivity in HgBa_{2}CaCu_{2}O_{6}, HgBa_{2}Ca_{2}Cu_{3}O_{8}, and Tl_{2}Ba_{2}CaCu_{2}O_{8} thin films have been investigated as functions of the magnetic field up to 18 T. We observe the universal scaling behavior between \rho_{xy} and \rho_{xx} in the regions of the clean and the moderately clean limit. The scaling exponent \beta is 1.9 in the clean limit at high field and low temperature whereas \beta is 1.0 in the moderately clean limit at low field and high temperature, consistent with a theory based on the midgap states in the vortex cores. This finding implies that the Hall conductivity is also universal in Hg- and Tl-based superconductors.
Universal scaling for the jamming transition  [PDF]
Michio Otsuki,Hisao Hayakawa
Physics , 2008, DOI: 10.1143/PTP.121.647
Abstract: The existence of universal scaling in the vicinity of the jamming transition of sheared granular materials is predicted by a phenomenology. The critical exponents are explicitly determined, which are independent of the spatial dimension. The validity of the theory is verified by the molecular dynamics simulation.
Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure  [PDF]
Pavol Quittner
Mathematics , 2014, DOI: 10.1007/s00208-015-1219-7
Abstract: We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation $$ u_t-\Delta u=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by nonlinear boundary conditions of the form $\partial u/\partial\nu=u^q$. Here $\nu$ denotes the outer unit normal on the boundary of the halfspace ${\mathbb R}^n_+$ and the exponents $p,q>1$ satisfy $p2$ (or $p<(n+2)/(n-2)$ and $q2$ and some symmetry of the solutions is assumed). As a typical application of our nonexistence results we provide optimal universal estimates for positive solutions of related problems in bounded and unbounded domains.
Scaling limits for exploration algorithms  [PDF]
Paola Bermolen,Matthieu Jonckheere,Jaron Sanders
Mathematics , 2015,
Abstract: We consider an exploration algorithm where at each step, a random number of items become active while related items get explored. Given an initial number of items $N$ growing to infinity and building on a strong homogeneity assumption, we study using scaling limits of Markovian processes statistical properties of the proportion of active nodes in time. This is a companion paper that rigorously establishes the claims and heuristics presented in [5]. [5] Jaron Sanders, Matthieu Jonckheere, and Servaas Kokkelmans. Sub-Poissonian statistics of jamming limits in Rydberg gases. 2015. To appear.
Universal Limits on Computation  [PDF]
Lawrence M. Krauss,Glenn D. Starkman
Physics , 2004,
Abstract: The physical limits to computation have been under active scrutiny over the past decade or two, as theoretical investigations of the possible impact of quantum mechanical processes on computing have begun to make contact with realizable experimental configurations. We demonstrate here that the observed acceleration of the Universe can produce a universal limit on the total amount of information that can be stored and processed in the future, putting an ultimate limit on future technology for any civilization, including a time-limit on Moore's Law. The limits we derive are stringent, and include the possibilities that the computing performed is either distributed or local. A careful consideration of the effect of horizons on information processing is necessary for this analysis, which suggests that the total amount of information that can be processed by any observer is significantly less than the Hawking-Bekenstein entropy associated with the existence of an event horizon in an accelerating universe.
On the scaling limits of planar percolation  [PDF]
Stanislav Smirnov,Oded Schramm
Mathematics , 2011, DOI: 10.1214/11-AOP659
Abstract: We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.
Universal scaling in sports ranking  [PDF]
Weibing Deng,Wei Li,Xu Cai,Alain Bulou,Qiuping A. Wang
Physics , 2011, DOI: 10.1088/1367-2630/14/9/093038
Abstract: Ranking is a ubiquitous phenomenon in the human society. By clicking the web pages of Forbes, you may find all kinds of rankings, such as world's most powerful people, world's richest people, top-paid tennis stars, and so on and so forth. Herewith, we study a specific kind, sports ranking systems in which players' scores and prize money are calculated based on their performances in attending various tournaments. A typical example is tennis. It is found that the distributions of both scores and prize money follow universal power laws, with exponents nearly identical for most sports fields. In order to understand the origin of this universal scaling we focus on the tennis ranking systems. By checking the data we find that, for any pair of players, the probability that the higher-ranked player will top the lower-ranked opponent is proportional to the rank difference between the pair. Such a dependence can be well fitted to a sigmoidal function. By using this feature, we propose a simple toy model which can simulate the competition of players in different tournaments. The simulations yield results consistent with the empirical findings. Extensive studies indicate the model is robust with respect to the modifications of the minor parts.
On scaling limits and Brownian interlacements  [PDF]
Alain-Sol Sznitman
Mathematics , 2012, DOI: 10.1007/s00574-013-0025-7
Abstract: We consider continuous time interlacements on Z^d, with d bigger or equal to 3, and investigate the scaling limit of their occupation times. In a suitable regime, referred to as the constant intensity regime, this brings Brownian interlacements on R^d into play, whereas in the high intensity regime the Gaussian free field shows up instead. We also investigate the scaling limit of the isomorphism theorem of arXiv:1111.4818. As a by-product, when d=3, we obtain an isomorphism theorem for Brownian interlacements.
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