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Stable laws and domains of attraction in free probability theory, With an appendix by Philippe Biane
 Ann. of Math
, 1999
"... with an appendix by Philippe Biane In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws ..."
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Cited by 27 (0 self)
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with an appendix by Philippe Biane In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated. 1.
Percolation transition in the Bose gas
"... The canonical partition function of a Bose gas gives rise to a probability distribution over the permutations of N particles. We study the probability and mean value of the cycle lengths in the cyclic permutations, their relation to physical quantities like pair correlations, and their thermodynamic ..."
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Cited by 20 (0 self)
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The canonical partition function of a Bose gas gives rise to a probability distribution over the permutations of N particles. We study the probability and mean value of the cycle lengths in the cyclic permutations, their relation to physical quantities like pair correlations, and their thermodynamic limit. We show that in the ground state of most interacting boson gas the mean cycle length diverges in the bulk limit and the particles form macroscopic cycles. In the free Bose gas BoseEinstein condensation is accompanied by a percolation transition: the appearance of infinite cycles with nonvanishing probability. To appear in J. Phys. A Permanent address: Central Research Institute for Physics, Budapest 1 Introduction This paper presents a new approach to phase transitions in bosonic systems. Since this description emerges somewhat accidentally from a study of the ferromagnetism in the Hubbard model, it may be interesting to outline the sequence of ideas connecting these seemingly ...
Introduction to Ergodic Theory of Chaotic Billards
 Pub. Mat. Rio de Janeiro: IMPA
, 2001
"... persing billiards, Ergodic Theory & Dynam. Systems 19 (1999), 201226. [Sz1] D. Szasz, Boltzmann's ergodic hypothesis, a conjecture for centuries ?, Studia Sci. Math. Hung. 31 (1996), 299322. [Sz2] D. Szasz, Hard ball systems and the Lorentz gas, Edited by D. Szasz. Springer, Berlin (2000). [Ta1] ..."
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Cited by 19 (5 self)
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persing billiards, Ergodic Theory & Dynam. Systems 19 (1999), 201226. [Sz1] D. Szasz, Boltzmann's ergodic hypothesis, a conjecture for centuries ?, Studia Sci. Math. Hung. 31 (1996), 299322. [Sz2] D. Szasz, Hard ball systems and the Lorentz gas, Edited by D. Szasz. Springer, Berlin (2000). [Ta1] S. Tabachnikov, Billiards. Panor. Synth. No. 1, SMF, Paris (1995). [Ta2] S. Tabachnikov, Exact transverse line fields and projective billiards in a ball, Geom. Funct. Anal. 7 (1997), 594608. [Ta3] S. Tabachnikov, Introducing projective billiards, Ergodic Theory & Dynam. Systems 17 (1997), 957976. [Va] L . N. Vaserstein, On Systems of particles with finite  range and/or repulsive interactions, Commun. Math. Phys. 69 (1979), 3156. [Vi] M. Viana, Stochastic dynamics of deterministic systems
RECTANGULAR RANDOM MATRICES, RELATED CONVOLUTION
, 2008
"... We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all non commutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in ..."
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Cited by 14 (9 self)
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We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all non commutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore, we can define a “rectangular free convolution”, which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values. This convolution is linearized by cumulants and by an analytic integral transform, that we called the “rectangular Rtransform”.
Incomplete markets over an infinite horizon: Longlived securities and speculative bubbles
 JOURNAL OF MATHEMATICAL ECONOMICS
, 1996
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Numerical inversion of multidimensional Laplace transforms by the Laguerre method
 Eval
, 1998
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Evolution and Extinction Dynamics in Rugged Fitness Landscapes.
, 1998
"... After an introductory section summarizing the paleontological data and some of their theoretical descriptions, we describe the `reset' model and its (in part analytically soluble) mean field version, which have been briefly introduced in Letters[1, 2]. Macroevolution is considered as a problem of ..."
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Cited by 8 (2 self)
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After an introductory section summarizing the paleontological data and some of their theoretical descriptions, we describe the `reset' model and its (in part analytically soluble) mean field version, which have been briefly introduced in Letters[1, 2]. Macroevolution is considered as a problem of stochastic dynamics in a system with many competing agents. Evolutionary events (speciations and extinctions) are triggered by fitness records found by random exploration of the agents' fitness landscapes. As a consequence, the average fitness in the system increases logarithmically with time, while the rate of extinction steadily decreases. This nonstationary dynamics is studied by numerical simulations and, in a simpler mean field version, analytically. We also consider the effect of externally added `mass' extinctions. The predictions for various quantities of paleontological interest (lifetime distribution, distribution of event sizes and behavior of the rate of extinction) a...
Hidden Symmetries of the Principal Chiral Model Unveiled
"... . By relating the twodimensional U(N) Principal Chiral Model to a simple linear system we obtain a freefield parametrisation of solutions. Obvious symmetry transformations on the freefield data give symmetries of the model. In this way all known `hidden symmetries' and Backlund transformations, a ..."
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Cited by 5 (3 self)
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. By relating the twodimensional U(N) Principal Chiral Model to a simple linear system we obtain a freefield parametrisation of solutions. Obvious symmetry transformations on the freefield data give symmetries of the model. In this way all known `hidden symmetries' and Backlund transformations, as well as a host of new symmetries, arise. 1 Introduction The definition of complete integrability for field theories remains rather imprecise. One usually looks for structures analogous to those existing in completely integrable hamiltonian systems with finitely many degrees of freedom, such as a Laxpair representation or conserved quantities equal in number to the number of degrees of freedom. A very transparent notion of integrability is that completely integrable nonlinear systems are actually simple linear systems in disguise. For example, the Inverse Scattering Transform for two dimensional integrable systems such as the KdV equation establishes a correspondence between the nonlinea...