One-dimensional random walk with self-interaction

A standard random walk on a one-dimensional integer lattice is considered where the probability ofk self-intersections of a path =(0, (1),..., (n) is proportional toe ^–k . It is proven that for <0,n ^–1/3(n) converges to a certain continuous random variable. For >0 the formulas are given for the asymptotic Westerwater velocity of a generic path and for the variance of the fluctuations about the asymptotic motion.     

On the existence of invariant,absolutely continuous measures

Let (, , ) be a measure space with normalized measure,f: a nonsingular transformation. We prove: there exists anf-invariant normalized measure which is absolutely continuous with respect to if and only if there exist >0, and , 0<<1, such that (E)< implies (f ^–k(E))< for allk0.     

Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy

We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain TT _c , H0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang–Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose extended analyticity; roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated extended dispersion relation.     

Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

Current distributions in a two-dimensional random-resistor network

The current and logarithm-of-the-current distributionsn(i) andn(ln i) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (–9k8) of n(ln i) i.e., ln i&^k, is found to scale with the linear sizeL as (InL)^(k). The exponents (k) are not inconsistent with the recent theoretical prediction (k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)–, yielding information on the threshold below which the multifractality of (i) breaks down. Our numerical results for the moments of the currents are consistent with other available results.     

Directed paths on percolation clusters

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is geometrical in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponents and, defined by ft and xt , describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard random energy exponents (=1/3 and =2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponents=1/2 and =v/v, where v and v are the directed percolation exponents. In addition, we predict the absence of a free phase in any dimension at the percolation threshold.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Modified London model for the determination of λ and ζ in a high κ superconductor

We present a modified London model suggested by Brandt [1–3] which introduces a finite vortex core size appropriate for isotropic superconductors in which the average internal field is less than approximately (1/4)H _c2. TheSR lineshape resulting from this model possesses a distinctive shape due to the magnetic penetration depth and the vortex core diameter (approximately equal to twice the coherence length ). However, for a given lineshape, there is a large range of values of and which produce nearly the same lineshape. Lineshape smearing caused by disorder in the vortex lattice increases uncertainty in values for and . If well-determined values of either (T) or (T) are not available from another technique, both of them can be determined bySR measurements alone if runs in more than one applied field at the same temperature are fit with and as shared parameters. We also present our method of estimating the degree of disorder in the vortex lattice.     

A Monomial Basis for the Virasoro Minimal Series M(p,p′) : The Case 1<p′/p<2

Quadratic relations are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The first case is the (2,1) primary fields for the (p,p)-minimal series M_r,s (1rp–1,1sp–1) for the Virasoro algebra where 1<p/p<2. We restrict ourselves to the case p3, for which the (2,1) primary field exists. The second case is the intertwiners corresponding to the two-dimensional representation for the level k integrable highest weight modules V() (0k) for the affine Lie algebra      

A solution spectrum of the nonlinear Schrödinger equation. II

It was shown in a previous communication that the nonlinear Schrödinger equation exhibits a spectrum of eigenfunctions of the form = _k,A _k (coshkx) ^–k and = _k B _k (coshkx) ^–k–1sinhkx, and the corresponding eigenvalues of the energy are related to a band structure with a characteristic energy gap as a significant feature. In the present paper, it is shown that a further spectrum exists exhibiting the general structure = _k=0 A _k(cosh kx)^–k–1/2and = _k=0 B_k(cosh kx)^–k–3/2sinhkx and yielding also a band structure. An extension of the solution spectrum to a nonlinear Klein-Gordon equation and a nonlinear Dirac equation does not imply essential difficulties, and the corresponding characteristic band structure has to be related to a mass spectrum.     

On ruelle's ξ-function for exponentially decreasing potentials

We show that Ruelle's generalised -function for a classical one-dimensional lattice spin system with two-body interaction (i) exp(-i )a(i) with >1 extends to a meromorphic function in the whole complex plane.     

A model of Piron's preparation-question structures in Ludwig's selection structures

We give a model of the basic Jauch-Piron (JP) approach to quantum physics, i.e., of preparation-question structure (with four basic axioms and without axioms C, P, A), in terms of Ludwig's selection structure; in the latter structure the primitive notion of individual sample of a physical entity is formally described (without making reference to any probability concept). Once we interpret Piron's concept of question in Ludwig's context of a selection structure, we find that there is no difficulty in formalizing notions such as performable together questions; moreover, results such as = or ()= can be formally proved. We develop the theory along the lines of the JP approach; the set of JP propositions is derived and it turns out to be a complete lattice, as happens in Piron's theory, but with a different physical interpretation of the lattice operations. Finally, we study some connections between the standard Ludwig foundation and our approach.     

A derivation of the Broadwell equation

We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e.N, 0,N²const, whereN is the number of particles and is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.Research partially supported by CNR-PS-MMAIT     

Estimates of general Mayer graphs. II. Long-range behavior of graphs with two root points occurring in the theory of ionized systems

We find the asymptotic behavior of general Mayer 2-graphs (Mayer graphs with two root points), which occur in the theory of ionized systems. This problem arises when one wants to compute corrections to the Debye length for large values of the plasma parameter. For a given 2-graph (r) with Debye-Hückel linese ^– /r, we prove the inequalitiesC _m r ^– e ^– (r) (r ₀)C_Mr^3k–l e ^– , for anyrr ₀, and whereC _m andC _M are positive and finite constants which depend only on . These bounds are finite whenever (r) is not infinite everywhere. The integersl, k, and denote, respectively, the number of lines of the graph , its number of field points, and its local line connectivity (the maximum number of chains linking the root points, which have no line in common). From this result, we deduce that the simple irreducible 2-graphs dominant at large distances decay exponentially likee ^– and have an isthmus between the root points (an isthmus is a line whose deletion separates the graph into two disjoint components, each one containing a root point). We prove also that 2-graphs that have a number of linesl > 3k+ are infinite. We exhibit simple, irreducible prototypes satisfying this condition, for anyk 6. This implies that the Abe-Meeron theory of ionized gases as applied to a classical plasma is not free from divergences. Finally, we extend the preceding results to 2-graphs with lines F_L=(e ^– /r)^k L, withk _L real positive. We prove that they still decay exponentially likee ^– , where is now the maximal flow in a network associated to by assigning the capacityk _L to each lineL.     

The lattice covering time problem for sites visitedk-times

The problem of the covering time for sites visitedk-times is defined as the mean time taken by a random walk to visit each site of a lattice at leastk times. We performed the investigation using Monte Carlo simulations over one dimensional lattices, ofN sites, with periodic boundary conditions. Two different regions are investigated:Nk1 andkN1. In the former region, we obtain a behaviour of the typet _k/t ₁=a –Bk ^–0.35+A(k)N ^–0.75, (a <1.6). In the latter region we obtain two possible behaviours:t _k k ^0.95 andt _k k(lnk)^–0.5. Two formulas which have a very close behaviour.     

Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A ₀ in a Hilbert space ₀ are defined formally as A ₍₎=A ₀+GG ^*, where G is an injective linear mapping from = ^d to the scale space _-k(A₀)k , kN, of generalized elements associated with the self-adjoint operator A ₀, and where is a self-adjoint operator in . The cases k=1 and k=2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k=2n>1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A ₍₎ in the general setting ran G – _k (A ₀), kN, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.     

More surprises in the general theory of lattice systems

I use Israel's methods to prove new theorems of ubiquitous pathology for classical and quantum lattice systems. The main result is the following: Let be any interaction and be any translation-invariant equilibrium state for (extremal or not). Then there exists a sequence { _k } of interactions converging to , having extremal (or even unique) translation-invariant equilibrium states _k , such that { _k } converges to . In certain situations the perturbations _k – can be chosen to lie in a cone of antiferromagnetic pair interactions. I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r ² interaction (Thouless effect).     

Spectral duality for planar billiards

For a bounded open domain with connected complement inR ² and piecewise smooth boundary, we consider the Dirichlet Laplacian- on and the S-matrix on the complement ^c . We show that the on-shell S-matricesS _k have eigenvalues converging to 1 askk ₀ exactly when-- has an eigenvalue at energyk ₀ ² . This includes multiplicities, and proves a weak form of transparency atk=k ₀. We also show that stronger forms of transparency, such asS _{k 0} having an eigenvalue 1 are not expected to hold in general.     

Averaged electron Green function and CDW pinning in one-dimensional random potential

The averaged retarded electron Green functionG ⁺(,k) in 1d disordered metal is calculated using the Berezinsky diagram technique. Using the Gorkov's theory it is shown, that the substitution of inG ⁺ (,k) by the square of the external frequency atk=0 gives the dependence of Fröhlich conductivity _F(). This dependence describes the impurity pinning of CDW in 1d disordered metals. The good agreement of this dependence with experimental data Zeller et al. about _F() in quasi-1d conductor KCP is found     

Some problems of quantum mechanics possessing a non-negative phase-space distribution function

In order to clarify physical consequences due to the presence of a set of auxiliary functions _k (q,t) in quantum mechanics with a non-negative phase-space distribution function, the simplest quantum-mechanical problems are solved. It is shown that _k (q,t) influence upon the results of a problem. Therefore it is supposed that _k (q, t) reflect some physical reality (subquantum situation), interacting with a mechanical system. In particular the subquantum situation determines the minimum coordinate and momentum uncertainties ((q)² and (p)²) as well as the coordinate distribution of a fixed system and the momentum distribution of a free system. These results provide the opportunity to formulate the notion of a stationary homogeneous isotropic subquantum situation. Supposing thatq andp are small an attempt is made to develop an approximate method of solutions (quasi-orthodox approximation). Energy spectrum of an electron in a hydrogen atom is found in the second order of this approximation.On leave of absence from Peoples' Friendship University, Chair of Theoretical Physics, 3, Ordjonikidze Street, B-302, Moscow, U.S.S.R.