Finite-size effects in random energy models and in the problem of polymers in a random medium

By the use of traveling wave equations we calculate the finite-size corrections to the free energy of random energy models in their low-temperature phases and in the neighborhood of the transition temperature. We find that although the extensive part of the free energy does not show any critical behavior when the temperature approaches its transition value, the finite-size corrections signal the transition by becoming singular. We obtain a scaling form for these finite-size corrections valid in the limitN andTT _c . By considering a generalized random energy model in the limit of a very large number of steps, we obtain results for the finite-size corrections in the problem of a polymer in a random medium.     

On the existence of thermodynamics for the random energy model

Derrida's random energy model is considered. Almost sure andL ^P convergence of the free energy at any inverse temperature are proven. Rigorous upper and lower bounds to the finite size corrections to the free energy are given.     

Finite size effects for the Ising model on random graphs with varying dilution

We investigate the finite size corrections to the equilibrium magnetization of an Ising model on a random graph with N nodes and N^γ edges, with 1<γ≤2. By conveniently rescaling the coupling constant, the free energy is made extensive. As expected, the system displays a phase transition of the mean-field type for all the considered values of γ at the transition temperature of the fully connected Curie-Weiss model. Finite size corrections are investigated for different values of the parameter γ, using two different approaches: a replica based finite N expansion, and a cavity method. Numerical simulations are compared with theoretical predictions. The cavity based analysis is shown to agree better with numerics.     

Fluctuation of inverse compressibility for electronic systems with random capacitive matrices

Abstract

This article is concerned with the statistics of the addition spectra of certain many-body systems of identical particles. In the first part, the pertinent system consists of N identical particles distributed among K<N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it with random coefficients. On a large scale, the ground-state energy E(N) of the whole system grows quadratically with N, but in general there is no simple relation such as E_N = aN+bN ². The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) X_N ≡E(N+1)?2E(N)+E(N?1) is a fluctuating quantity. Regarding the numbers X_N as values assumed by a certain random variable X, we obtain a closed-form expression for its distribution F _(X). Its main feature is that the corresponding density P _(X)=dF _(X)/d _X has a maximum at the point X=0. As K→∞ the density is Poissonian, namely, P(X)→e^?X

This result serves as a starting point for the second part, in which coupling between subsystems is included. More generally, a classical model is suggested in order to study fluctuations of Coulomb blockade peak spacings in large two-dimensional semiconductor quantum dots. It is based on the electrostatics of several electron islands among which there are random inductive and capacitive couplings. Each island can accommodate electrons on quantum orbitals whose energy depends also on an external magnetic field. In contrast to a single-island quantum dot, where the spacing distribution between conductance peaks is close to Gaussian, here the distribution has a peak at small spacing value. The fluctuations are mainly due to charging effects. The model can explain the occasional occurrence of couples or even triples of closely spaced Coulomb blockade peaks, as well as the qualitative behaviour of peak positions with the applied magnetic field.     

Diluted generalized random energy model

A layered random spin model, equivalent to the generalized random energy model (GREM), is introduced. In analogy with diluted spin systems, a diluted GREM (DGREM) is constructed. It can be applied to calculate approximately the thermodynamic properties of spin glass models in low dimensions. For the Edwards-Anderson model it gives the correct critical dimension and 5% accuracy for the ground state energy in two dimensions. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 6, 415–419 (25 March 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Exact energy eigenvalues by summation of thejwkb series

For the potentialV(x)=V ₀ tan² x, the corrections to the lowest orderjwkb (Bohr-Sommerfeld) energy quantization rule are non-zero. These higher order corrections are explicitly computed using the formalism of Dunham. The resultingjwkb series for the energy eigenvalues is summable, and yields the exact bound state spectrum.     

Are random walks random?

Random walks have been created using the pseudo-random generators in different computer language compilers (BASIC, PASCAL, FORTRAN, C++) using a Pentium processor. All the obtained paths have apparently a random behavior for short walks (2¹⁴ steps). From long random walks (2³³ steps) different periods have been found, the shortest being 2¹⁸ for PASCAL and the longest 2³¹ for FORTRAN and C++, while BASIC had a 2²⁴ steps period. The BASIC, PASCAL and FORTRAN long walks had even (2 or 4) symmetries. The C++ walk systematically roams away from the origin. Using deviations from the mean-distance rule for random walks, d² ∝ N, a more severe criterion is found, e.g. random walks generated by a PASCAL compiler fulfills this criterion to N < 10 000.     

Metastable states of spin glasses on random thin graphs

In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution . Thin graphs are graphs where the local connectivity of each site is fixed to some value c. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as c in agreement with previous studies showing that finite connectivity corrections of order 1/c increase the number of metastable states with respect to the totally connected mean field limit. We also prove that the average number of metastable states in the limit is finite and converges to the average number of metastable states in the Sherrington-Kirkpatrick model. An annealed calculation for the number of metastable states of energy E is also carried out giving a lower bound on the ground state energy of these spin glasses. For small c one may obtain analytic expressions for . Received 14 October 1999 and Received in final form 14 December 1999     

Quasi-long-range order in the random anisotropy Heisenberg model

The random field and random anisotropy N-vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg (N=3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈m(r ₁)m(r ₂)〉∼|r ₁−r ₂|^{− 0.62ε}. The magnetic susceptibility diverges at low fields as χ∼H ^−1+0.15ε. In the random field N-vector model the correlation length is finite at arbitrarily weak disorder for any N>3. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 2, 130–135 (25 July 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Anderson impurity model: Vertex corrections within the finite U non-crossing approximation

We revise the simplest possible approximations to solve numerically the vertex equations for the single impurity Anderson model (SIAM) within the finite U non-crossing approximation (UNCA), considering the self-energies at lowest order in the 1/N diagrammatic expansion. We introduce an approximation to the vertex corrections that includes the double energy dependence and compare it with an approximation (NCAf²v) that neglects a second energy argument. Finally, we analyse the influence of the different approximations on the estimated Kondo scale for simple electronic models.     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

Mixed spin transverse Ising model with longitudinal random crystal field interactions

The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory. In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators. We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice (N=3) and a simple-cubic lattice (N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features resulting from the fluctuation of the crystal field interactions. Received: 13 February 1998 / Accepted: 17 March 1998     

Transmission coefficient and heat conduction of a harmonic chain with random masses: Asymptotic estimates on products of random matrices

We find upper and lower bounds for the transmission coefficient of a chain of random masses. Using these bounds we show that the heat conduction in such a chain does not obey Fourier's law: For different temperatures at the ends of a chain containingN particles the energy flux falls off likeN ^–1/2 rather thanN ^–1.Research supported by a ZWO fellowship and in part by U.S.A.F.O.S.R. grant no. 78-3522     

The thermodynamics of the Curie-Weiss model with random couplings

We study the Curie-Weiss version of an Ising spin system with random, positively biased couplings. In particular, the case where the couplings _ij take the values one with probabilityp and zero with probability 1 –p, which describes the Ising model on a random graph, is considered. We prove that ifp is allowed to decrease with the system sizeN in such a way thatNp(N) asN , then the free energy converges (after trivial rescaling) to that of the standard Curie-Weiss model, almost surely. Similarly, the induced measures on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed.     

Vibrational energy transport in molecular wires

Motivated by recent experimental observation (see, e.g., I. V. Rubtsov, Acc. Chem. Res. 42, 1385 (2009)) of vibrational energy transport in (CH₂O) _N and (CF₂) _N molecular chains (N = 4–12), in this paper we present and solve analytically a simple one dimensional model to describe theoretically these data. To mimic multiple conformations of the molecular chains, our model includes random off-diagonal couplings between neigh-boring sites. For the sake of simplicity, we assume Gaussian distribution with dispersion σ for these coupling matrix elements. Within the model we find that initially locally excited vibrational state can propagate along the chain. However, the propagation is neither ballistic nor diffusion like. The time T _m for the first passage of the excitation along the chain, scales linearly with N in the agreement with the experimental data. Distribution of the excitation energies over the chain fragments (sites in the model) remains random, and the vibrational energy, transported to the chain end at t = T _m is dramatically decreased when σ is larger than characteristic interlevel spacing in the chain vibrational spectrum. We do believe that the problem we have solved is not only of intellectual interest (or to rationalize mentioned above experimental data) but also of relevance to design optimal molecular wires providing fast energy transport in various chemical and biological reactions.     

Characteristic Polynomials¶of Real Symmetric Random Matrices

The correlation functions of the random variables det(λ−X), in which X is an hermitian N×N random matrix, are known to exhibit universal local statistics in the large N limit. We study here the correlation of those same random variables for real symmetric matrices (GOE). The derivation relies on an exact dual representation of the problem: the k-point functions are expressed in terms of finite integrals over (quaternionic) k×k matrices. However the control of the Dyson limit, in which the distance of the various parameters λ's is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson–Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a finite number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large N limit. Received: 19 March 2001 / Accepted: 21 June 2001     

Poisson point processes,cascades, and random coverings ofR n

The generalized random energy model (GREM) is formulated in terms of hierarchies of Poisson point processes. This allows one to relate the high-temperature region with a random covering ofR ⁿ .     

Field theory of the random flux model

The random flux model (defined here as a model of lattice fermions hopping under the influence of maximally random link disorder) is analysed field theoretically. It is shown that the long range physics of the model is described by the supersymmetric version of a field theory that has been derived earlier in connection with lattice fermions subject to weak random hopping. More precisely, the field theory relevant for the behaviour of n-point correlation functions is of non-linear σ model type, where the group GL(n|n) is the global invariant manifold. It is argued that the model universally describes the long range physics of random phase fermions and provides further evidence in favour of the existence of delocalised states in the middle of the band in two dimensions. The same formalism is applied to the study of non-Abelian generalisations of the random flux model, i.e. N-component fermions whose hopping is mediated by random U(N) matrices. We discuss some physical applications of these models and argue that, for sufficiently large N, the existence of long range correlations in the band centre (equivalent to metallic behaviour in the Abelian case) can be safely deduced from the RG analysis of the model.     

Two-dimensional random walk on the ground state energy surface: The N-Z distributions from nuclear heavy-ion collisions

The nucleon transfer process in deep inelastic collision is formulated as a two-dimensional random walk process on the N-Z plane. The probability of each step is assumed to be proportional to the available final state density which is determined by the ground state energy surface of the colliding dinuclear system. Here a model problem is considered, where the long and narrow valley in the energy surface is replaced by a slot with a flat bottom. Even in this crude model, good agreement is obtained between the calculated widths of the N-Z distribution and the experimental values, indicating that the ground state energy surface plays an important role in the mass and charge transfer phenomena in deep inelastic processes.     

Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ ^∞-norm of the corresponding eigenvectors is of order O(N ^−1/2), modulo logarithmic corrections. The upper bound O(N ^−1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK. Partially supported by NSF grant DMS-0602038.