Exact solution of a drop-push model for percolation

Motivated by a computer science algorithm known as "linear probing with hashing," we study a new type of percolation model whose basic features include a sequential "dropping" of particles on a substrate followed by their transport via a "pushing" mechanism. Our exact solution in one dimension shows that, unlike the ordinary random percolation model, the drop-push model has nontrivial spatial correlations generated by the dynamics itself. The critical exponents in the drop-push model are also different from those of the ordinary percolation. The relevance of our results to computer science is pointed out.     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

Exact solution of the Faddeev-Volkov model

The Faddeev-Volkov model is an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. It serves as a lattice analog of the sinh-Gordon and Liouville models and intimately connected with the modular double of the quantum group Uq(sl₂). The free energy of the model is exactly calculated in the thermodynamic limit. In the quasi-classical limit c→+∞ the model describes quantum fluctuations of discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In the strongly-coupled limit c→1 the model turns into a discrete version of the D=2 Zamolodchikov's “fishing-net” model.     

Exact solution of a deterministic model for isomerization kinetics

Isomerization kinetics is studied on a one-dimensional ideal gas model with deterministic transitions. The concentrations of species are found to satisfy the phenomenological rate laws appropriate for diffusion-controlled kinetics, and the various correlations are determined. In the long-time regime, higher correlations present long tails reflecting a strongly non-Markovian evolution.     

Exact solution of phantom dark energy model

We investigate the phantom dark energy model derived from the scalar field with a negative kinetic term.By assuming a particular relation between the time derivative of the phantom field and the Hubble function,an exact solution of the model is constructed.Absence of the ’big rip’ singularity is shown explicitly.We then derive special features of phantom dark energy model and show that its predictions are consistent with all astrophysical observations.     

Exact operator solution of the Calogero-Sutherland model

The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.     

Exact solution of the simplified hubbard model

We present the exact solution of the simplified Hubbard model in which only one kind of electrons can hop and this quantum mechanical hopping of electrons is assumed to be unconstrained. It is shown that the model still behaves nontrivially, although it no longer depends on the lattice structure and the dimensionality of the system. For this case we find: (i) a gap in the ground state energy always exists at the half-filled band point (n=1), (ii) a preferred magnetic state atn=1 and largeU is a total spin singlet, (iii)U-dependence of the ground state energy has qualitatively the same form as one of the conventional Hubbard model with the (t ²/U)-behavior at largeU. A phase diagram of the model is discussed.     

Exact solution for the thermo Jaynes--Cummings model

Based on the construction of supersymmetric generators, we use the Lewis--Riesenfeld invariant method to deduce the exact and explicit eigen-energy spectrum with the time-dependent thermo Jaynes--Cummings model. One of the advantages of this approach is that it can transform the hidden form, related to the chronological product, of the time evolution operator into an explicit expression. Moreover, the dynamical and statistics properties of physical quantities are obtained for the given initial states in the thermo Jaynes--Cummings system.     

Exact solution of a triangular ising model in a nonzero magnetic field

A closed-form expression is obtained for the free energy per site of the Ising model on the triangular lattice in a nonzero magnetic field and with two- and three-site interactions. The solution is valid along a trajectory in the parameter space, and is derived using a method of exact decimation. A criterion determining the validity of the decimation method is also established.     

Exact solution of a modified periodic Anderson model in one dimension

A periodic Anderson model for electron correlation in a localized-electron band mixed with a conduction band is exactly solved in one dimension, by a Bethe Ansatz, in a modified version with interaction only between right- or left-going wavenumbers. The singlet-ground-state energy for the symmetric model is obtained for the half-filled case N/N_a=2.     

Exact solution of the general non-intersecting string model

We present a thorough analysis of the non-intersecting string (NIS) model and its exact solution. This is an integrable q-states vertex model describing configurations of non-intersecting polygons on the lattice. The exact eigenvalues of the transfer matrix are found by the analytic Bethe ansatz. The Bethe ansatz equations thus found are shown to be equivalent to those for a mixed spin model involving both and infinite spin. This indicates that the NIS model provides a representation of the quantum group corresponding to spins and s = ∞. The partition function and the excitations in the thermodynamic limit are computed.     

Exact solution of an evolutionary model without aging

We introduce an age-structured asexual population model containing all the relevant features of evolutionary aging theories. Beneficial as well as deleterious mutations, heredity, and arbitrary fecundity are present and managed by natural selection. An exact solution without aging is found. We show that fertility is associated with generalized forms of the Fibonacci sequence, while mutations and natural selection are merged into an integral equation which is solved by Fourier series. Average survival probabilities and Malthusian growth exponents are calculated and indicate that the system may exhibit mutational meltdown. The relevance of the model in the context of fissile reproduction groups like many protozoa and coelenterates is discussed.     

Exact solution of a repulsive fermi model with enhanced superconducting correlations

We present the exact solution of a model of interacting fermions in any dimension with a pure repulsive interaction projecting out a given Cooper channel. The solution rests upon the infinite ranged character of the interaction in real space, leading to a functional integral that is dominated by a Gaussian term. The solution produces strong superconducting enhancements and quasi-long-ranged order in a channel that is not present in the Hamiltonian explicitly, but of the form given by arguments from order by projection.     

Exact solution of a model DNA-inversion genetic switch with orientational control

DNA inversion is an important mechanism by which bacteria and bacteriophage switch reversibly between phenotypic states. In such switches, the orientation of a short DNA element is flipped by a site-specific recombinase enzyme. We propose a simple model for a DNA-inversion switch in which recombinase production is dependent on the switch state (orientational control). Our model is inspired by the fim switch in E. coli. We present an exact analytical solution of the chemical master equation for the model switch, as well as stochastic simulations. Orientational control causes the switch to deviate from Poissonian behavior: the distribution of times in the on state shows a peak and successive flip times are correlated.