antiferromagnetic potts model in a magnetic field a finite size scaling study abstract the finite size scaling method is used to investigate the phase diagrams of the q state antiferromagnetic potts model in an external magnetic field on a square lattice critical lines where the system undergoes second order phase transitions between ordered and disordered phases are found which are in complete qualitative agreement with known results for q 3 also new unexpected features are found for q 4 where the transition vanishes in the negative field region while in the positive field region as q increases the critical line flattens out with increasing slopes both at the t 0 h h c point where h c is the critical field at zero temperature and at the t 0 h 0 point thermal exponents y t are also calculated for arbitrary q along the critical lines

on the convergence rates of genetic algorithms this paper discusses the convergence rates of genetic algorithms by using the minorization condition in the markov chain theory we classify genetic algorithms into two kinds one with time invariant genetic operators another with time variant genetic operators for the former case we have obtained the bound on its convergence rate on the general state space for the later case we have bounded its convergence rate on the finite state space

the thermodynamic limit on bethe lattices the author emphasises the fact that to obtain the exact solution of hamiltonian models on bethe lattices one can apply explicitly the formal method of rigorous statistical mechanics i e the thermodynamic limit of probability measures this approach solves the well known dichotomy between clashing alternative solutions and provides a very simple analytic solution for a large class of hamiltonian models

some ab percolation problems in the antiferromagnetic potts model abstract the ab correlated site random bond percolation problem in a q state antiferromagnetic potts model on bethe lattices is solved we find the analytic expression of the ab percolation characteristic functions in terms of the temperature the external field and the active bond concentration p b the ab threshold and the phase boundary of the system coincide at zero temperature and at most in two other points for every constant p b 1 the properties of the bethe lattice allow us to find the temperature dependent p b which defines the ab droplets i e those special ab clusters which diverge with thermal exponents along the phase boundary

critical behaviour in three state potts antiferromagnets on a bethe lattice abstract a rigorous method recently introduced for the solution of spin models on bethe lattices is used to solve the three state antiferromagnetic potts model in an external field and to find its phase diagram critical lines where the system undergoes first order or second order phase transitions are found which are in complete qualitative agreement with recent results by racz and vicsek also new unexpected ordered phases are found

phase diagrams of the q state potts model on bethe lattices abstract we use a rigorous recently developed method of solution of spin models on bethe lattices cayley trees it allows us to detect all the phases of the ferromagnetic and antiferromagnetic q state potts models in a field fpm and apm respectively the phase diagrams of the fpm are characterized by three critical lines of first order transitions in the apm the ordered phases which appear always for h 0 and under certain conditions for h 0 and or h

statistical dynamics of the royal road genetic algorithm abstract metastability is a common phenomenon many evolutionary processes both natural and artificial alternate between periods of stasis and brief periods of rapid change in their behavior in this paper an analytical model for the dynamics of a mutation only genetic algorithm ga is introduced that identifies a new and general mechanism causing metastability in evolutionary dynamics the ga s population dynamics is described in terms of flows in the space of fitness distributions the trajectories through fitness distribution space are derived in closed form in the limit of infinite populations we then show how finite populations induce metastability even in regions where fitness does not exhibit a local optimum in particular the model predicts the occurrence of fitness epochs periods of stasis in population fitness distributions at finite population size and identifies the locations of these fitness epochs with the flow s hyperbolic fixed points this enables exact predictions of the metastable fitness distributions during the fitness epochs as well as giving insight into the nature of the periods of stasis and the innovations between them all these results are obtained as closed form expressions in terms of the ga s parameters an analysis of the jacobian matrices in the neighborhood of an epoch s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space s topological structure more general quantitative features of the dynamics fitness fluctuation amplitudes epoch stability and speed of the innovations are also determined from the jacobian eigenvalues the analysis shows how quantitative predictions for a range of dynamical behaviors that are specific to the finite population dynamics can be derived from the solution of the infinite population dynamics the theoretical predictions are shown to agree very well with statistics from ga simulations we also discuss the connections of our results with those from population genetics and molecular evolution theory

global bethe lattice consideration of the spin 1 ising model the spin 1 ising model with bilinear and biquadratic exchange interactions and single ion crystal field is solved on the bethe lattice using exact recursion equations the general procedure of investigation of critical properties is discussed and a full set of phase diagrams are constructed for both positive and negative biquadratic couplings a comparison is made with the results of other approximation schemes

new order parameters in the potts model on a cayley tree for the q state potts model new order parameters projecting on a group of spins instead of a single spin are introduced on a cayley tree this allows the physical interpretation of the potts model at noninteger values q0 of the number of states the model can be solved recursively this recursion exhibits chaotic behaviour changing qualitatively at critical values of q0 using an additional order parameter belonging to a group of zero extrapolated size the additional ordering is related to a percolation problem this percolation distinguishes different phases and explains the critical indices of percolation class occuring at the peierls temperature

potts models on feynman diagrams we investigate numerically and analytically potts models on thin random graphs generic feynman diagrams using the idea that such models may be expressed as the limit of a matrix model the thin random graphs in this limit are locally tree like in distinction to the fat random graphs that appear in the planar feynman diagram limit more familiar from discretized models of two dimensional gravity the interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree like structures such as the bethe lattice q state potts models display a first order transition in the mean field for q 2 so the thin graph potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation such discontinuous transitions also appear in multiple ising models on thin graphs and may have implications for the use of the replica trick in spin glass models on random graphs

drift analysis and average time complexity of evolutionary algorithms the computational time complexity is an important topic in the theory of evolutionary algorithms eas this paper reports some new results on the average time complexity of eas based on drift analysis some useful drift conditions for deriving the time complexity of eas are studied including conditions under which an ea will take no more than polynomial time in problem size to solve a problem and conditions under which an ea will take at least exponential time in problem size to solve a problem the paper first presents the general results and then uses several problems as examples to illustrate how these general results can be applied to concrete problems in analyzing the average time complexity of eas while previous work only considered 1 1 eas without any crossover the eas considered in this paper are fairly general which use a finite population crossover mutation and selection 2001 elsevier science b v all rights reserved

description of the translation invariant splitting gibbs measures for the potts model on a cayley tree for the q state potts model on a cayley tree of order k ge 2 it is well known that at sufficiently low temperatures there are at least q 1 translation invariant gibbs measures which are also tree indexed markov chains such measures are called translation invariant splitting gibbs measures tisgms in this paper we find all tisgms and show in particular that at sufficiently low temperatures their number is 2 q 1 we prove that there are q 2 where a is the integer part of a critical temperatures at which the number of tisgms changes and give the exact number of tisgms for each intermediate temperature for the binary tree we give explicit formulae for the critical temperatures and the possible tisgms while we show that these measures are never convex combinations of each other the question which of these measures are extremals in the set of all gibbs measures will be treated in future work

probability measures and hamiltonian models on bethe lattices i properties and construction of mrt probability measures the properties of one step markov rotationally and m step m 1 or 2 translationally invariant mrt probability measures on q state site qss bethe lattices are studied a theorem is proven which completely defines such measures in terms of m q2 q fundamental probabilities these are explicitly calculated for any mrt qss hamiltonian model as a consequence of our approach the dychotomy between alternative solutions of hamiltonian models on bethe lattices is solved

cones of rank 2 and the poincare bendixson property for a new class of monotone systems abstract we study systems that are monotone in a generalized sense with respect to cones of rank 2 the main result of the paper is the existence of a poincare bendixson property for some solutions of those systems

potts model on a cayley tree and logistic equation the q state potts model on a cayley tree can be solved by a recursion formula depending on the properties at the surface the model on a bethe lattice is obtained by extrapolating the interior of a cayley tree sufficiently far from the surface in order to have a stable fixed point of the recursion for q 2 we find a second order transition of percolation type at the bethe peierls temperature and a first order transition at a higher temperature for coordination number z 3 the recursion extrapolated to q 1 is identical to the logistic equation the feigenbaum route to chaos appears for antiferromagnetic coupling of the potts model the first period doubling corresponds to a multicritical point in the phase diagram of the potts model

first order transitions in percolation models abstract we describe the analytic solution of a potts correlated site random bond percolation model which exhibits a first order percolative transition at the potts critical point the subject is discussed in connection with the droplet theory in lattice spin models

equivalence of ferromagnetic spin models on trees and random graphs recently we have found identical behaviour for various spin models on thin random graphs feynman diagrams and the corresponding bethe lattices in this paper we observe that the ratios of the saddle point equations in the random graph approach are identical to the fixed point s of the recursion relations which are used to solve the models on the bethe lattice the loops in the random graphs thus have no influence on the thermodynamic limit for such ferromagnetic spin models we consider the correspondence explicitly for ising and q state potts models and also note that multispin interaction models on cacti admit a similar correspondence with a randomized version of the cacti graphs which contain loops

existence of periodic orbits for high dimensional autonomous systems abstract we give a result on existence of periodic orbits for autonomous differential systems with arbitrary finite dimension it is based on a poincare bendixson property enjoyed by a new class of monotone systems introduced in l a sanchez cones of rank 2 and the poincare bendixson property for a new class of monotone systems j differential equations 216 2009 1170 1190 a concrete application is done to a scalar differential equation of order 4

a new approach to estimating the expected first hitting time of evolutionary algorithms evolutionary algorithms ea have been shown to be very effective in solving practical problems yet many important theoretical issues of them are not clear the expected first hitting time is one of the most important theoretical issues of evolutionary algorithms since it implies the average computational time complexity in this paper we establish a bridge between the expected first hitting time and another important theoretical issue i e convergence rate through this bridge we propose a new general approach to estimating the expected first hitting time using this approach we analyze eas with different configurations including three mutation operators with without population a recombination operator and a time variant mutation operator on a hard problem the results show that the proposed approach is helpful for analyzing a broad range of evolutionary algorithms moreover we give an explanation of what makes a problem hard to eas and based on the recognition we prove the hardness of a general problem

how to analyse evolutionary algorithms many variants of evolutionary algorithms have been designed and applied the experimental knowledge is immense the rigorous analysis of evolutionary algorithms is difficult but such a theory can help to understand design and teach evolutionary algorithms in this survey first the history of attempts to analyse evolutionary algorithms is described and then new methods for continuous as well as discrete search spaces are presented and discussed

a potts ising correspondence on thin graphs we note that it is possible to construct a bond vertex model that displays q state potts criticality on an ensemble of 3 random graphs of arbitrary topology which we denote as thin random graphs in contrast to the fat graphs of the planar diagram expansion since the four vertex model in question also serves to describe the critical behaviour of the ising model in field the formulation reveals an isomorphism between the potts and ising models on thin random graphs on planar graphs a similar correspondence is present only for q 1 the value associated with percolation

probability measures and hamiltonian models on bethe lattices ii the solution of thermal and configurational problems in a previous paper we introduced a method for the construction of rotationally and translationally invariant probability measures generated by one step markov hamiltonian models on q state site bethe lattices here the corresponding thermal problems are solved by finding the relative free energy which gives complete information on the properties of the models under study configurational problems also can be solved with the present tools as an example the solution of polychromatic correlated site random bond percolation models is found