×

zbMATH — the first resource for mathematics

Brownian-motion ensembles of random matrix theory: a classification scheme and an integral transform method. (English) Zbl 1215.82039
Summary: We study a class of Brownian-motion ensembles obtained from the general theory of Markovian stochastic processes in random-matrix theory. The ensembles admit a complete classification scheme based on a recent multivariable generalization of classical orthogonal polynomials and are closely related to Hamiltonians of Calogero-Sutherland-type quantum systems. An integral transform is proposed to evaluate the \(n\)-point correlation function for a large class of initial distribution functions. Applications of the classification scheme and of the integral transform to concrete physical systems are presented in detail.

MSC:
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
33C90 Applications of hypergeometric functions
81R12 Groups and algebras in quantum theory and relations with integrable systems
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Okolowicz, J.; Ploszajczak, M.; Rotter, I., Phys. rep., 374, 271, (2003)
[2] Ferry, D.K.; Goodnick, S.M., Transport in nanostructures, (1997), Cambridge Univ. Press Cambridge
[3] Stöckmann, H.-J., Quantum chaos: an introduction, (2000), Cambridge Univ. Press Cambridge
[4] Zirnbauer, M.R., J. math. phys., 37, 4986, (1996)
[5] Helgason, S., Differential geometry, Lie groups and symmetric spaces, (1978), Academic New York · Zbl 0451.53038
[6] Dyson, F.J.; Dyson, F.J., J. math. phys., J. math. phys., 3, 1199, (1962)
[7] Dyson, F.J.; Oppermann, R.; Wegner, F., Phys. rev., Z. phys. B, 34, 327, (1979)
[8] Altland, A.; Zirnbauer, M.R., Phys. rev. B, 55, 1142, (1997)
[9] Balian, R., Nuovo cimento B, 57, 183, (1968)
[10] Caselle, M.; Magnea, U., Phys. rep., 394, 41, (2004)
[11] Baranger, H.U.; Mello, P.A.; Jalabert, R.A.; Pichard, J.-L.; Beenakker, C.W.J., Phys. rev. lett., Europhys. lett., 27, 255, (1994)
[12] Risken, H., The fokker – planck equation, (1989), Springer · Zbl 0665.60084
[13] Caselle, M., unpublished
[14] Helgason, S., Geometric analysis on symmetric spaces, (1994), American Mathematical Society · Zbl 0809.53057
[15] Baker, T.H.; Forrester, P.J., Commun. math. phys., 188, 175, (1997)
[16] Mehta, M.L., Random matrices, (2004), Elsevier Amsterdam · Zbl 0594.60067
[17] Katori, M.; Tanemura, H., J. math. phys., 45, 3058, (2004)
[18] Dyson, F.J., J. math. phys., 3, 1191, (1962)
[19] Lenz, G.; Haake, F., Phys. rev. lett., 65, 2325, (1990)
[20] Beenakker, C.W., Rev. mod. phys., 69, 731, (1997)
[21] Macêdo, A.M.S., Phys. rev. B, 61, 4453, (2000)
[22] Dennery, P.; Krzywicki, A., Mathematics for physicists, (1971), Dover New York · Zbl 0156.46201
[23] Adler, M.; Forrester, P.J.; Nagao, T.; van Moerbeke, P., J. stat. phys., 99, 141, (2000)
[24] Gardiner, G.W., Handbook of stochastic methods, (1985), Springer
[25] Ha, Z.N.C., Phys. rev. lett., 73, 1574, (1994)
[26] Ha, Z.N.C., Nucl. phys. B, 435, 604, (1995)
[27] Lesage, F.; Pasquier, V.; Serban, D., Nucl. phys. B, 435, 585, (1995)
[28] Serban, D.; Lesage, F.; Pasquier, V., Nucl. phys. B, 466, 499, (1996)
[29] Gurappa, N.; Panigrahi, P.K., Phys. rev. B, 62, 1943, (2000)
[30] Gurappa, N.; Panigrahi, P.K.; Shreecharan, T., J. comput. appl. math., 160, 103, (2003)
[31] Gurappa, N.; Panigrahi, P.K., unpublished
[32] Stanley, R.P., Adv. math., 77, 76, (1989)
[33] Sogo, K., J. phys. soc. jpn., 65, 3097, (1996)
[34] Lassalle, M., C. R. acad. sci. Paris, ser. I, 312, 425, (1991)
[35] Dorokhov, O.N.; Mello, P.A.; Pereyra, P.; Kumar, N., JETP lett., Ann. phys. (N.Y.), 181, 290, (1988)
[36] Titov, M.; Brouwer, P.W.; Furusaki, A.; Mudry, C., Phys. rev. B, 63, 235318, (2001)
[37] Imamura, T.; Wadati, M., J. phys. soc. jpn., 71, 1511, (2002)
[38] French, J.B.; Kota, V.K.B.; Pandey, A.; Tomsovic, S., Ann. phys. (N.Y.), 181, 198, (1988)
[39] Pandey, A., Chaos solitons fractals, 5, 1275, (1995)
[40] Beenakker, C.W.J.; Rejaei, B., Phys. rev. B, 49, 7499, (1994)
[41] Fyodorov, Y.V., Nucl. phys. B, 621, 643, (2002)
[42] Fyodorov, Y.V.; Strahov, E., J. phys. A: math. gen., 36, 3203, (2003)
[43] Brezin, E.; Hikami, S., Commun. math. phys., 214, 111, (2000)
[44] Brezin, E.; Hikami, S., J. phys. A: math. gen., 36, 711, (2003)
[45] Keating, J.P.; Snaith, N.C., Commun. math. phys., 214, 57, (2000)
[46] Andreev, A.V.; Simons, B.D., Phys. rev. lett., 75, 2304, (1995)
[47] Verbaarschot, J.J.M.; Wetting, T., Annu. rev. nucl. part. sci., 50, 343, (2000)
[48] Kaneko, J., SIAM J. math. anal., 24, 1086, (1993)
[49] Guhr, T., J. math. phys., 32, 336, (1991)
[50] Guhr, T., Ann. phys. (N.Y.), 250, 145, (1996)
[51] Brouwer, P.W.; Frahm, K., Phys. rev. B, 53, 1490, (1996)
[52] Efetov, K.B., Supersymmetry in disorder and chaos, (1997), Cambridge Univ. Press Cambridge · Zbl 0990.82501
[53] Strahov, E.; Fyodorov, Y.V., Commun. math. phys., 241, 343, (2003)
[54] Fyodorov, Y.V.; Strahov, E., Nucl. phys. B, 630, 343, (2002)
[55] Tsvelik, A.M., unpublised
[56] Desrosiers, P.; Lapointe, L.; Mathieu, P., Czechoslovak J. phys., 54, 1223, (2004)
[57] Desrosiers, P.; Lapointe, L.; Mathieu, P., Nucl. phys. B, 606, 547, (2001)
[58] Sergeev, A.N.; Veselov, A.P., Commun. math. phys., 245, 249, (2004)
[59] Sergeev, A.N., Theor. math. phys., 131, 747, (2002)
[60] Macedo-Junior, A.F.; Macêdo, A.M.S., Phys. rev. B, 66, 041307(R), (2002)
[61] Mello, P.A., Phys. rev. lett., 60, 1089, (1988)
[62] Brouwer, P.W.; Beenakker, C.W.J., J. math. phys. (N.Y.), 37, 4904, (1996)
[63] Datta, S., Introduction to mesoscopic physics, (1995), Cambridge Univ. Press Cambridge
[64] Braun, P.; Heusler, S.; Müller, S.; Haake, F., unpublised
[65] Savin, D.V.; Sommers, H.-J., Phys. rev. B, 73, 08137(R), (2006)
[66] Blanter, Ya.M.; Büttiker, M., Phys. rep., 336, 2, (2000)
[67] Mello, P.A., J. phys. A: math. gen., 23, 4061, (1990)
[68] Heusler, S.; Müller, S.; Braun, P.; Haake, F., Phys. rev. lett., 96, 066804, (2006)
[69] Dyson, F.J., J. math. phys., 13, 90, (1972)
[70] Mikhlin, S.G., Integral equations, (1964), Pergamon New York · Zbl 0117.31902
[71] Beenakker, C.W.J., Nucl. phys. B, 422, 515, (1994)
[72] Frahm, K.; Frahm, K., Phys. rev. lett., J. phys. I (France), 5, 877, (1995)
[73] Bateman, H., The higher transcendental functions, vol. I, (1953), McGraw-Hill New York, pp. 174-175
[74] Samuel, S., J. math. phys., 21, 2695, (1980)
[75] Mello, P.A.; Baranger, H.U., Waves random media, 9, 105, (1999)
[76] Beenakker, C.W.J., Phys. rev. B, 46, 12841, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.