Random chains: exact solutions and exact results

 

The topic of random chains was my thesis subject; I took it up from lectures by Marc Kac, who contrasted this with the problem of spin glasses, on which I worked much later.
The so-called Dyson-Schmidt equation expresses that a semi-infinite random chain remains invariant when one random element is added. This equation was first generalized to complex frequencies, [P1].
Then a whole class of exactly solvable systems was discovered, which gave work for years because of the may different applications: random alloy and tight-binding models [P3], harmonic and liquid metal models [P5], a certain random variable [P8], Ising chains in a random magnetic field [P12], their correlation functions [P24], more general random field Ising models [P28], site-dilution [L1], the presence of short range order, [L7], and some universal fluctuations [L15].

Paper [P4] deals with a moment expansion for the density of states of random chains, used to determine their specific heat.

A related quasi-periodic chain was also investigated, [L3].

Other works listed here on random one-dimensional systems are [P29], [P27], [P20], [P18], [P17], [P14], [P10], [P6].

 

[L15] Th.M. N. and M.C.W. van Rossum, Universal fluctuations in a simple disordered system , Phys. Lett. A 160, (1991) 461-464

[L7] Th.M. N., Exact solutions for one-dimensional systems with short range order,
Europhys. Lett. 4 (1987) 1109-1114

[L3] J.M. Luck and Th.M. N., A soluble quasi-crystalline magnetic model: the XY quantum spin chain, Europhys. Lett. 2 (1986) 257-266

[L1] Th.M. N., Exactly soluble diluted random one-dimensional lattices,
Phys. Lett. A 103 (1984) 333-336

[P29] Th.M. N., Singularies in spectra of disordered systems, Physica A 167 (1990) 43-65

[P28] M. Funke, Th.M. N., and S. Trimper, Exact solution for Ising chains in a random field,
J. Phys. A 22 (1989) 5097-5107

[P27] Th.M. N. and H. Brandt, Diffusion and survival in a medium with imperfect traps,
J. Stat. Phys. 59 (1990) 53-72

[P24] J.M. Luck and Th.M. N., Correlation function of random field Ising chains: is Lorentzian or not?,
J. Phys. A 22 (1989) 2151-2180 (Elisabeth Gardner Memorial Issue)

[P20] J.M. Luck and Th.M. N., Lifshitz tails and long-time decay in random systems with arbitrary disorder, J. Stat. Phys. 52 (1988) 1-22

[P19] Th.M. N. and J.M. Luck, Lifshitz singularies in the total and the wavenumber dependent spectral density of random harmonic chains,
Physica A 145 (1987) 161-189

[P18] Th.M. N. and J.M. Luck, Lifshitz singularies in random harmonic chains: Periodic amplitudes near the band edge and near special frequencies,
J. Stat. Phys. 48 (1987) 393-424

[P14] Th.M. N., J.M. Luck, J. Canisius, J.L. van Hemmen, and W.J.Ventevogel, Special frequencies and Lifshitz singularies in binary random chains,
J. Stat. Phys. 45 (1986) 395-417

[P12] Th.M. N. and J.M. Luck, Exactly soluble random field Ising models in one dimension,
J. Phys. A 19 (1986) 1207-1227

[P10] Th.M. N. and J.M. Luck, Singular behavior of the density of states and the Lyapunov coefficient in binary random harmonic chains,
J. Stat. Phys. 41 (1985) 745-771

[P8] C. de Calan, J.M. Luck, Th.M. N., and D. Petritis, On the distribution of a random variable occurring in one-dimensional disordered systems,
J. Phys. A 18 (1985) 501-523

[P6] Th.M. N. and M.H. Ernst, Transport and spectral properties of strongly disordered chains, Phys. Rev. B 31 (1985) 3518-3533

[P5] Th.M. N., Exact solutions for spectra and Greens functions in one-dimensional random systems, Physica A 125 (1984) 197-236

[P4] Th.M. N., Low frequency expansion and specific heat for harmonic chains with random masses, J. Phys. A 17 (1984) 1111-1121

[P3] Th.M. N., Exact electronic spectra and inverse localization lengths in one-dimensional random systems, Physica A 120 (1983) 468-514

[P1] Th.M. N., A new approach to the problem of disordered harmonic chains,
Physica A 113, (1982) 173-202