cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

random permutations of countable sets the main objective of this paper is a study of random decompositions of random point configurations onr d into finite clusters this is achieved by constructing for each configurationz a random permutation ofz with finite cycles these cycles then form the cluster decomposition ofz it is argued that a good candidate for a random permutation ofz is a gibbs measure for a certain specification and conditions are given for the existence and uniqueness of such a gibbs measure these conditions are then verified for certain random configurationsz

finite cycle gibbs measures on permutations of mathbb z d we consider gibbs distributions on the set of permutations of mathbb z d associated to the hamiltonian h sigma sum x v sigma x x where sigma is a permutation and v mathbb z d rightarrow mathbb r is a strictly convex potential call finite cycle those permutations composed by finite cycles only we give conditions on v ensuring that for large enough temperature alpha 0 there exists a unique infinite volume ergodic gibbs measure mu alpha concentrating mass on finite cycle permutations this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions we construct mu alpha as the unique invariant measure of a markov process on the set of finite cycle permutations that can be seen as a loss network a continuous time birth and death process of cycles interacting by exclusion an approach proposed by fernandez ferrari and garcia define tau v as the shift permutation tau v x x v in the gaussian case v vert cdot vert 2 we show that for each v in mathbb z d mu alpha v given by mu alpha v f mu alpha f tau v cdot is an ergodic gibbs measure equal to the thermodynamic limit of the specifications with tau v boundary conditions for a general potential v we prove the existence of gibbs measures mu alpha v when alpha is bigger than some v dependent value

lattice permutations and poisson dirichlet distribution of cycle lengths we study random spatial permutations on 3 where each jump x x is penalized by a factor mathrm e t x pi x 2 the system is known to exhibit a phase transition for low enough t where macroscopic cycles appear we observe that the lengths of such cycles are distributed according to poisson dirichlet this can be explained heuristically using a stochastic coagulation fragmentation process for long cycles which is supported by numerical data

spatial random permutations and infinite cycles we consider systems of spatial random permutations where permutations are weighed according to the point locations infinite cycles are present at high densities the critical density is given by an exact expression we discuss the relation between the model of spatial permutations and the ideal and interacting quantum bose gas

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

loop correlations in random wire models we introduce a family of loop soup models on the hypercubic lattice the models involve links on the edges and random pairings of the link endpoints on the sites we conjecture that loop correlations of distant points are given by poisson dirichlet correlations in dimensions three and higher we prove that in a specific random wire model that is related to the classical xy spin system the probability that distant sites form an even partition is given by the poisson dirichlet counterpart

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

conformal invariance of loops in the double dimer model the dimer model is the study of random dimer covers perfect matchings of a graph a double dimer configuration on a graph mathcal g is a union of two dimer covers of mathcal g we introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes relative to a fixed set of faces of loops in the double dimer model on a planar graph as an application we prove that in the scaling limit of the uniform double dimer model on mathbb z 2 or on any other bipartite planar graph conformally approximating mathbb c the loops are conformally invariant

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

link lengths and their growth powers for a certain infinite family of knots or links we study the growth power ratios of their stick number lattice stick number minimum lattice length and minimum ropelength compared with their minimum crossing number c k for every it is known that the stick number and lattice stick number grow between the and linear power of the crossing number and minimum lattice length and minimum ropelength grow with at least the power of crossing number which is called the four thirds power law furthermore the minimal lattice length and minimum ropelength grow at most as o but it is unknown whether any family exhibits superlinear growth for any real number r between and 1 we give an infinite family of non splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number furthermore for any real number r between and 1 we give another infinite family of non splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number

shift in critical temperature for random spatial permutations with cycle weights we examine a phase transition in a model of random spatial permutations which originates in a study of the interacting bose gas permutations are weighted according to point positions the low temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of bose einstein condensates in our simplified model point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as ewens type weights on cycle lengths of permutations the critical temperature of the transition to long cycles depends on an interaction strength parameter for weak interactions the shift in critical temperature is expected to be linear in with constant of linearity c using markov chain monte carlo methods and finite size scaling we find c 0 618 0 086 this finding matches a similar analytical result of ueltschi and betz we also examine the mean longest cycle length as a fraction of the number of sites in long cycles recovering an earlier result of shepp and lloyd for non spatial permutations

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

on a model of random cycles we consider a model of random permutations of the sites of the cubic lattice permutations are weighted so that sites are preferably sent onto neighbors we present numerical evidence for the occurrence of a transition to a phase with infinite macroscopic cycles

loss network representation of peierls contours we present a probabilistic approach for the study of systems with exclusions in the regime traditionally studied via cluster expansion methods in this paper we focus on its application for the gases of peierls contours found in the study of the ising model at low temperatures but most of the results are general we realize the equilibrium measure as the invariant measure of a loss network process whose existence is ensured by a subcriticality condition of a dominant branching process in this regime the approach yields besides existence and uniqueness of the measure properties such as exponential space convergence and mixing and a central limit theorem the loss network converges exponentially fast to the equilibrium measure without metastable traps this convergence is faster at low temperatures where it leads to the proof of an asymptotic poisson distribution of contours our results on the mixing properties of the measure are comparable to those obtained with duplicated variables expansion used to treat systems with disorder and coupled map lattices it works in a larger region of validity than usual cluster expansion formalisms and it is not tied to the analyticity of the pressure in fact it does not lead to any kind of expansion for the latter and the properties of the equilibrium measure are obtained without resorting to combinatorial or complexanalysis techniques

subharmonic solutions for nonautonomous sublinear second order hamiltonian systems some existence theorems are obtained for subharmonic solutions of nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

subharmonic solutions for second order differential systems ink this paper we consider the existence of subharmonic solutions for the problem utt g u t where g rn r is not necessarily convex and r rn is periodic with minimal period t 0

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

spatial random permutations with small cycle weights we consider the distribution of cycles in two models of random permutations that are related to one another in the first model cycles receive a weight that depends on their length the second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps we prove the occurrence of infinite macroscopic cycles above a certain critical density

best packing of identical helices in this paper we prove the unique existence of a ropelength minimizing conformation of the spun double helix in a mathematically rigorous way and find the minimal ropelength where t is the unique solution in of the equation using this result the pitch angles of the standard triple and quadruple helices are around and respectively which are almost identical with the approximated pitch angles of the zero twist structures previously known by olsen and bohr we also find the ropelength of the standard n helix

taming griffiths singularities infinite differentiability of quenched correlation functions we prove infinite differentiability of the magnetization and of all quenched correlation functions for disordered spin systems at high temperature or strong magnetic field in the presence of griffiths singularities we also show uniqueness of the gibbs state and exponential decay of truncated correlation functions with probability one our results are obtained through new simple modified high temperature or low activity expansions whose convergence can be displayed by elementary probabilistic arguments our results require no assumptions on the probability distributions of the random parameters except for the obvious one of no percolation of infinite couplings and in the strong field situation for the also obvious requirement that zero magnetic fields do not percolate

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

periodic and subharmonic solutions of a class of subquadratic second order hamiltonian systems some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

on the central limit theorem for stationary mixing random fields a simple proof of a central limit theorem for stationary random fields under mixing conditions is given generalizing some results obtained by more complicated methods e g bernstein s method

minimum lattice length and ropelength of 2 bridge knots and links knots are commonly found in molecular chains such as dna and proteins and they have been considered to be useful models for structural analysis of these molecules one interested quantity is the minimum number of monomers necessary to realize a molecular knot the minimum lattice length len k of a knot k indicates the minimum length necessary to construct k in the cubic lattice another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the complexity of knot conformation the minimum ropelength rop k is the minimum length of an ideally flexible rope necessary to tie a given knot k much effort has been invested in the research project for finding upper bounds on both quantities in terms of the minimum crossing number c k of the knot it is known that len k and rop k lie between o c k 34 and o c k ln c k 5 but unknown yet whether any family of knots has superlinear growth in this paper we focus on 2 bridge knots and links linear growth

random permutations of a regular lattice spatial random permutations were originally studied due to their connections to bose einstein condensation but they possess many interesting properties of their own for random permutations of a regular lattice with periodic boundary conditions we prove existence of the infinite volume limit under fairly weak assumptions when the dimension of the lattice is two we give numerical evidence of a kosterlitz thouless transition and of long cycles having an almost sure fractal dimension in the scaling limit finally we comment on possible connections to schramm lowner curves