Random Processes  

Randomness in Mathematics and in General > s.a. matrices; probability.
@ General references: Shale JFA(79) [functions]; Compagner AJP(91) [definitions]; Bennett 98 [history]; Eagle BJPS(05) [as unpredictability].
@ Random numbers: Vattulainen et al PRL(94) [tests]; Stefanov et al qp/99 [generators, using quantum mechanics].

Random Process > s.a. computing; Fiber [fiber process]; fractal; stochastic processes.
* Idea: (The most) random (infinite) sequences are those which cannot be described by algorithms shorter than the sequances themselves (Bernoulli chain/sequence).
* Examples: No computable number is random (see ).
* Remark: It is impossible to prove that any given sequence was generated by a random process.
@ References: Ochs Nat(90)jan, Yockey Nat(90)apr; Chaitin NS(90)mar.
> Related topics: see statistical geometry [including Poisson and other point processes].

Random Walk > s.a. diffusion.
* Idea: A probability measure on the space of all paths on a space; For example, Brownian motion.
* Specification: Can be assigned as a transition probability from each incomplete, n-step path, to each extension to n+1 steps; A simple type of situation is a Markov process, in which the probability only depends on the n-th configuration.
* Displacement: In a space of Hausdorff dim d, the mean square displacement after N steps is D2 N 2/d; The planar (d = 2) self-avoiding walk has rms displacement exponent 3/4 (long conjectured, proved in 2001).
* Applications: Used to sample a space of states in randomized algorithms.
@ General references: Feynman I-6-6 (simple); Franceschetti et al AJP(93) [as eigenvalue problem]; in Ambjørn et al 97; Marchetti & da Silva BJP(99)mp/04 [and Brownian motion]; Barkema et al PRL(01) [with random static traps]; Campanino & Petritis in(04)m.PR/02 [survey and relevance]; Ferraro & Zaninetti PhyA(04) [statistics of visits]; Nakamura & Small PLA(07) [test].
@ On random sets: Barat & Chakrabarti PRP(95); Faggionato et al CMP(06); Caputo & Faggionato m.PR/06; Zeitouni JPA(06) [rev]; Bogachev a0707-in [rev]; Juhász JPA(08) [renormalization group approach]; > s.a. voronoi tilings.
@ On other sets: Bender et al JMP(94) [non-integer dimension]; Woess 00 [infinite graphs]; Davison et al JPA(01) [fractals]; Kostrykin & Schrader m.CO/04 [on graphs, generating functions]; Franceschetti JSP(07) [inside n-disk].
@ Quantum: Aharonov et al PRA(93) [introduction]; Childs et al QIP(02)qp/01 [vs classical]; Dür et al PRA(02)qp [experiment proposal]; Kempe CP(03)qp [rev]; Martin et al PRD(05)gq/04 [in generalized quantum mechanics]; Konno FNL(05)qp/04 [quantum to classical, path integral]; Kosik CEJP(03) [two models]; Strauch PRA(06) [discrete- and continuous-time], JMP(07) [and Dirac equation, relativistic effects]; Wang & Douglas qp/07 [applied to graph isomorphism problem]; Yin et al a0708 [in random environment]; Manouchehri & Wang JPA(07) [continuous time, discrete space]; > s.a. classical limit, decoherence, graph, quantum computation.
@ Quantum, 1D: Carteret et al JPA(03)qp [asymptotics]; Romanelli et al PhyA(04)qp/03 [as Markov process]; Fuss et al a0705 [analytic solution].
@ Quantum, 2D: Watabe et al a0802 [1-parameter family, limit distributions].
@ Continuous time: Manouchehri & Wang qp/06/PRA [state space must be discrete]; Jafarizadeh & Sufiani a0704 [spectral analysis].
@ Self-avoiding: Prellberg JPA(01) [scaling]; Hueter m.PR/01 [D = 2], m.PR/01 [D > 2].
@ Related topics: Bellissard et al JPA(97) [distributions, using non-commutative geometry]; Zia & Schmittmann AJP(03) [distribution of variances].

Other Random Systems > s.a. chaos; ising models; lattice field theory; quantum klein-gordon fields; scattering.
@ General references: Grandy FP(92) [thermodynamics]; Svozil 93; Mézard cm/95-ln [replica field theory]; Drozdz & Wojcik nt/00 [emergence of order]; Mulhall et al PRL(00)nt [randomly interacting spins]; Donetti & Destri JPA(04) [scale-free random trees].
@ Random medium: Lima et al PRL(01) [deterministic walk]; > s.a. quantum particle, Transport, waves.
@ In dynamics: Ornstein 74; Gaspard & Wang PRP(93); Johnson et al PRL(98) [and orderly spectra]; Clifford & Stirzaker PRS(08) [history-dependent].
@ In quantum mechanics: Yurtsever qp/98 [algorithmic randomness]; Ulfbeck & Bohr FP(01) ["genuine fortuitousness"]; de la Torre a0707.
> Geometry, combinatorics: see dynamical triangulations [including surfaces], regge calculus; types of graphs.

Random Fields > s.a. electricity [random dipoles].
@ General references: Bertschinger ApJS(01)ap [multiscale simulation package]; de Dominicis & Giardina 06 [r JSP(07)].
@ Related topics: Vojta JPA(97), JPA(97) [damage spreading, Ising model].

Other Related Issues
@ Interpolation of randomly sampled data: Lombardi & Schneider A&A(02)ap [in astronomy].

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