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We study sequences of random variables obtained by iterative procedures, which can be thought of as nonlinear generalizations of the arithmetic mean. We prove a strong law of large numbers for a class of such iterations. This gives rise... more
We study sequences of random variables obtained by iterative procedures, which can be thought of as nonlinear generalizations of the arithmetic mean. We prove a strong law of large numbers for a class of such iterations. This gives rise to the concept of generalized expected value of a random variable, for which we prove an analog of the classical Jensen inequality. We give several applications to models arising in mathematical physics and other areas.
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Rigorous results concerning mathematical models of disordered systems in classical statistical mechanics are presented. They fall into two categories. Results of the first category are general and dimension-independent. They concern the... more
Rigorous results concerning mathematical models of disordered systems in classical statistical mechanics are presented. They fall into two categories. Results of the first category are general and dimension-independent. They concern the order of magnitude of fluctuations of extensive quantities; the notion of extensive quantity is introduced in Chapter II and plays a central role there. Results of the second category concern absence of certain phase transitions, caused by randomness (the rounding effect). They are more special and apply only in dimensions which are low enough. The proofs are based on probabilistic tools, such as martingales and moment generating functions.
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ABSTRACT A Reply to the Comment by R. Mannella and P. V. E. McClintock.
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Research Interests: Mathematics, Mathematical Physics, Physics, Quantum Physics, Charged Particle Optics, and 14 morePure Mathematics, Magnetic field, Dynamics, Diffusion, Mathematical Analysis, Brownian Motion, Approximation, Motion, Stochastic differential equation, Perturbations, Differential equation, THERMOPHORESIS, Limit (Mathematics), and integrals
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We show that the distribution of the percolation threshold in a large finite system does not converge to a Gaussian when the size of the system goes to infinity, provided that the two widely accepted definitions of correlation length are... more
We show that the distribution of the percolation threshold in a large finite system does not converge to a Gaussian when the size of the system goes to infinity, provided that the two widely accepted definitions of correlation length are equivalent. The shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length. The result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v. All results are rigorously proven in the 2D case. Generalizations for three dimensions are also discussed.
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Condensed matter physics often has to consider systems with static disorder [6], [7], i.e. with impurities, dislocations, substitutions etc. which vary from sample to sample (thus introducing disorder) but which do not exhibit thermal... more
Condensed matter physics often has to consider systems with static disorder [6], [7], i.e. with impurities, dislocations, substitutions etc. which vary from sample to sample (thus introducing disorder) but which do not exhibit thermal fluctuations on relevant time scales (hence the word static). To account for such disorder mathematically one often uses lattice spin systems with random parameters in the interaction (e.g. random magnetic fields or random coupling constants). For each fixed realization of these parameters one then obtains a spin system in which the usual quantities of physical interest — magnetization, free energy etc. — can be calculated. Random parameters of this type are often called quenched, to stress the fact that they remain constant during the calculation of spin averages — corresponding to the static nature of the disorder in the modelled physical system.
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Research Interests: Engineering, Mathematics, Biophysics, Economics, Biology, and 15 moreMedicine, Multidisciplinary, Molecular Dynamics, Noise, Correlation, Mathematical Analysis, Economic analysis, Article, Electrical Parameters, Electric Circuit, Nature Communications, Feedback System, Master Equation, Multiplicative noise, and Biological Analysis
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We analyze the disordered geometry resulting from random permutations of Euclidean space. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the... more
We analyze the disordered geometry resulting from random permutations of Euclidean space. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the perturbations, and provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. In Part II, we will use this to prove that a geodesic with random initial conditions is almost surely not minimizing. We also develop in this paper some general results on conditional Gaussian measures.
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We study central limit theorems for certain nonlinear sequences of random variables. In particular, we prove the central limit theorems for the bounded conductivity of the random resistor networks on hierarchical lattices.
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We continue our analysis of geodesics in quenched, random Riemannian environments. In this article, we prove that a geodesic with randomly chosen initial conditions is almost surely not minimizing. To do this, we show that a minimizing... more
We continue our analysis of geodesics in quenched, random Riemannian environments. In this article, we prove that a geodesic with randomly chosen initial conditions is almost surely not minimizing. To do this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.