Exact Potts Model Partition Functions for Strips of the Square Lattice

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L _t and for arbitrarily great length L , with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v)= . We give general formulas for N _{Z, G, j} and its specialization to v=–1 for arbitrary L _t for both types of boundary conditions, as well as other general structural results on Z. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus , arising as the accumulation set of partition function zeros as L , in the q plane for fixed v and in the v plane for fixed q.     

Exact Potts Model Partition Functions for Strips of the Triangular Lattice

We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)= $\sum _{j = 1}^{N_{Z,G,\lambda } }$ c _z,G,j (λ _z,G,j ) ^m-1. We give general formulas for N _Z,G,j and its specialization to v=?1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus ${\mathcal{B}}$ , arising as the accumulation set of partition function zeros as m→∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L=3 (free) and L=3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q=3 Potts antiferromagnet on the 2D triangular lattice is given.     

Quenched Averages for Self-Avoiding Walks and Polygons on Deterministic Fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages ,, and over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent are the same for the annealed and quenched averages. However, , and , where the exponents and , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives and , to be compared with the known annealed values and .     

Weakly Non-Ergodic Statistical Physics

For weakly non ergodic systems, the probability density function of a time average observable is where is the value of the observable when the system is in state j=1,…L. p _j ^eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _j ^eq is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x ²〉∼t ^α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered . We briefly discuss possible physical applications in single particle experiments.     

An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model

In terms of the loop algebra and the algebraic Bethe-ansatz method, we derive the invariant subspace associated with a given Ising-like spectrum consisting of 2 ^r eigenvalues of the diagonal-to-diagonal transfer matrix of the superintegrable chiral Potts (SCP) model with arbitrary inhomogeneous parameters. We show that every regular Bethe eigenstate of the τ ₂-model leads to an Ising-like spectrum and is an eigenvector of the SCP transfer matrix which is given by the product of two diagonal-to-diagonal transfer matrices with a constraint on the spectral parameters. We also show in a sector that the τ ₂-model commutes with the loop algebra, , and every regular Bethe state of the τ ₂-model is of highest weight. Thus, from physical assumptions such as the completeness of the Bethe ansatz, it follows in the sector that every regular Bethe state of the τ ₂-model generates an -degenerate eigenspace and it gives the invariant subspace, i.e. the direct sum of the eigenspaces associated with the Ising-like spectrum.     

Interfaces for Random Cluster Models

A random cluster measure on that is not translationally invariant is constructed for d3, the critical density p _c , and sufficiently large q. The resulting measure is proven to be a Gibbs state satisfying cluster model DLR- equations.     

Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as . For the planar Gaussian analytic function, , we show that this probability is asymptotic to . For the hyperbolic Gaussian analytic functions, , we show that this probability decays like .In the planar case, we also consider the problem posed by Mikhail Sodin² on moderate and very large deviations in a disk of radius r, as . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.Research supported by NSF grant #DMS-0104073 and NSF-FRG grant #DMS-0244479.     

Distribution of Particles Which Produces a “Smart” Material

If A _q(β, α, k) is the scattering amplitude, corresponding to a potential , where D⊂ℝ³ is a bounded domain, and is the incident plane wave, then we call the radiation pattern the function , where the unit vector α, the incident direction, is fixed, β is the unit vector in the direction of the scattered wave, and k>0, the wavenumber, is fixed. It is shown that any function , where S ² is the unit sphere in ℝ³, can be approximated with any desired accuracy by a radiation pattern: , where ∊ >0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ∊, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D _m⊂ D, 1≤ m≤ M, distributed in an a priori given bounded domain D⊂ℝ³. The geometrical shape of a small particle D _m is arbitrary, the boundary S _m of D _m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude , at a fixed k>0, arbitrarily close in the norm of L ²(S ²× S ²) to an arbitrary given scattering amplitude f(α ', α), corresponding to a real-valued potential q∊ L ²(D), i.e., corresponding to an arbitrary refraction coefficient in D. MSC: 35J05, 35J10, 70F10, 74J25, 81U40, 81V05, 35R30. PACS: 03.04.Kf.     

Multiple Intersection Exponents for Planar Brownian Motion

Let p≥2, n ₁≤⋅⋅⋅≤n _p be positive integers and be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a p-fold intersection exponent ς _p (n ₁,…,n _p ), as the exponential rate of decay of the probability that the packets , i=1,…,p, have no joint intersection. The case p=2 is well-known and, following two decades of numerical and mathematical activity, Lawler et al. (Acta Math. 187:275–308, 2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for p>2. We present an extensive mathematical and numerical study, leading to an exact formula in the case n ₁=1, n ₂=2, and several interesting conjectures for other cases.     

WZW Orientifolds and Finite Group Cohomology

The simplest orientifolds of the WZW models are obtained by gauging a symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion , where ζ is an element of the center of G. It reverses the sign of the Kalb-Ramond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in [31]. More generally, one may gauge orientifold symmetry groups that combine the -action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-Γ cohomology that we solve for all simple simply connected compact Lie groups G and all orientifold groups . Membre du C.N.R.S.     

Modules-at-Infinity for Quantum Vertex Algebras

This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian , denoted by DY _q (sl ₂) and with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra V _q and prove that every DY _q (sl ₂)-module is naturally a V _q -module. We also show that -modules are what we call V _q -modules-at-infinity. To achieve this goal, we study what we call -local subsets and quasi-local subsets of for any vector space W, and we prove that any -local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.     

Contribution of Sea Quarks and Gluons to the Proton Spin

Contributions of sea quark and gluon spins to the proton spin in Drell-Jahn processes and direct production of photons in proton-proton and proton-antiproton collisions are studied in the present work. Analytical expressions for two-spin asymmetries and are derived. In both processes, these asymmetries are studied and analyzed as functions of the kinematic variables , x _T, and x _F. Measurements of two-spin asymmetries and make it possible to determine the individual contributions of sea quark and gluon spins to the proton spin.     

The Bloch-Okounkov Correlation Functions of Classical Type

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc., and it can also be interpreted as correlation functions on integrable -modules of level one. Such -correlation functions at higher levels were then calculated by Cheng and Wang. In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of of type B, C, D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B, C, D and some fermionic type q-identities.     

A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL ^d lattice scales like . I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.     

Non-Lifshitz Tails at the Spectral Bottom of Some Random Operators

In this paper we continue with the investigation of the behavior of the integrated density of states of random operators of the form H _ω =−∇ ρ _ω ∇. In the present work we are interested in its behavior at 0, the bottom of the spectrum of H _ω . We prove that it converges exponentially fast to the integrated density of states of some periodic operator . Being periodic, cannot exhibit a Lifshitz behaviour. This result relates to the result of S.M. Kozlov (Russ. Math. Surv. 34(4):168–169, 1979) and improves it. Research partially supported by the Research Unity 01/UR/ 15-01 projects.     

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians , which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets (-valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.      

Anomalous Diffusion Index for Lévy Motions

In modelling complex systems as real diffusion processes it is common to analyse its diffusive regime through the study of approximating sequences of random walks. For the partial sums one considers the approximating sequence of processes . Then, under sufficient smoothness requirements we have the convergence to the desired diffusion, . A key assumption usually presumed is the finiteness of the second moment, and, hence the validity of the Central Limit Theorem. Under anomalous diffusive regime the asymptotic behavior of S _n may well be non-Gaussian and . Such random walks have been referred by physicists as Lévy motions or Lévy flights. In this work, we introduce an alternative notion to classify these regimes, the diffusion index . For some properly chosen let . Relationship between , the infinitesimal diffusion coefficients and the diffusion constant will be explored. Illustrative examples as well as estimates, based on extreme order statistics, for will also be presented.     

Single-pion production in pp collisions at 0.95 GeV/c (II)

The single-pion production reactions pp d , pp np and pp pp were measured at a beam momentum of 0.95GeV/c ( T _p 400 MeV) using the short version of the COSY-TOF spectrometer. The central calorimeter provided particle identification, energy determination and neutron detection in addition to time-of-flight and angle measurements from other detector parts. Thus all pion production channels were recorded with 1-4 overconstraints. The main emphasis is put on the presentation and discussion of the np channel, since the results on the other channels have already been published previously. The total and differential cross-sections obtained are compared to theoretical calculations. In contrast to the pp channel we observe in the np channel a strong influence of the excitation. In particular, the pion angular distribution exhibits a (3 cos² + 1) -dependence, typical for a pure s -channel excitation and identical to that observed in the d channel. Since the latter is understood by a s -channel resonance in the ¹ D ₂ pn partial wave, we discuss an analogous scenario for the pn channel.     

Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

We present exact results on the partition function of the q-state Potts model on various families of graphs G in a generalized external magnetic field that favors or disfavors spin values in a subset I _s ={1,…,s} of the total set of possible spin values, Z(G,q,s,v,w), where v and w are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial Ph(G,q,s,w) that counts the number of colorings of the vertices of G subject to the condition that colors of adjacent vertices are different, with a weighting w that favors or disfavors colors in the interval I _s . We derive powerful new upper and lower bounds on Z(G,q,s,v,w) for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for Z(G,q,s,v,w) on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.