monte carlo study of the point for collapsing trees we investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions two methods are used 1 a stochastic optimization process of the robbins monro type which is designed solely to locate the maximum value of the specific heat and 2 umbrella sampling which is designed to sample data over a wide temperature range as well as to combat the quasiergodicity of metropolis algorithms in the collapsed phase we find good evidence that the transition is second order with a divergent specific heat and that the divergence of the specific heat coincides with the metric collapse

adsorbing and collapsing directed animals a model of a self interacting directed animal which also interacts with a solid wall is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition the directed animal is confined to a 45 wedge and it interacts with one of the walls of this wedge the existence of a thermodynamic limit in this model shown and the presence of an adsorption transition is demonstrated by using constructive techniques by comparing this model to a process of directed percolation we show that there is also a collapse or transition in this model we examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity the generating function of adsorbing directed animals in a half space is found next from which we find the tricritical exponents associated with the adsorption transition a full solution for a collapsing directed animal seems intractible so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next

statistics of collapsing lattice animals we consider several lattice models of branched polymers in dilute solution in which the polymer molecule is modelled as a tree or animal in general the thermodynamic properties of the polymer are determined by monomer monomer and monomer solvent interactions we examine a two variable model which incorporates both types of interaction and discuss its relationship to other models which have previously been investigated in particular we discuss the collapse transition in these models

two dimensional lattice vesicles and polygons we consider a lattice model of two dimensional vesicles in which the boundary of the vesicle is the perimeter of a self avoiding polygon embeddable in the square lattice with fixed boundary length m we incorporate an osmotic pressure difference by associating a fugacity with the area enclosed by the polygon we derive rigorous results concerning the behaviour of the associated free energy and the form of the phase diagram by deriving exact values ofthe numbers of polygons with m edges which enclose area n and analysing the resulting series we obtain the free energy the phase boundary and various scaling exponents and amplitudes numerically

modelling the collapse transition in branched polymers abstract we consider a number of lattice animal models for the collapse transition exhibited by branched polymers in dilute solution in all models the collapse is driven by either some kind of near neighbour fugacity or cycle fugacity we describe rigorous upper and lower bounds on the temperature dependence of the limiting free energy and compare these bounds with numerical estimates derived using new exact enumeration data finally we comment on the difficulties associated with the interpretation of results for the specific heat

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

on the use of extrinsic probabilities in the computation of non bayesian cramer rao bounds for coded linearly modulated signals this contribution considers the non bayesian cramer rao bound crb related to parameter estimation from a linearly modulated signal observed in additive white gaussian noise we compare the exact crb expression for coded modulation with two ad hoc crb acrb expressions the first acrb is obtained by substituting in the exact crb expression for uncoded modulation the a priori symbol probabilities by the extrinsic symbol probabilities the second acrb which has received some attention in recent scientific publications additionally assumes that the real and imaginary parts of the symbols are independent our exposition focuses on the particular case of phase shift estimation by means of examples we show that although for some coded modulation schemes the exact and acrbs yield virtually the same numerical result for other coded modulation schemes the acrbs differ considerably among themselves and from the exact crb we provide some explanations for this behavior we also argue that both acrbs are expected to virtually coincide with the exact crb in the case of bit interleaved coded modulation combined with a rectangular constellations with independent in phase and quadrature mapping and a binary code for which the factor graph does not contain short cycles

1 d expansions for the free energy of weakly embedded site animal models of branched polymers we give the 1 d expansion to order 1 d5 for the limiting reduced free energy of a weakly embedded two variable site animal model of branched polymers hence we are able to derive expansions for the free energy of related one variable models and the growth constant of weakly and strongly embedded trees again to order 1 d5 we argue that the free energy expansions are asymptotically correct for a small range of fugacities only exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals especially for large d this result is not reflected in the free energy 1 d expansions where trees make the dominant contribution we also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1 d expansions thus we give partition functions for the general d dimensional hypercubic lattice for one to 13 sites

potts model formulation of branched polymers in a solvent a potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given the migdal kadanoff renormalisation group is employed to study the critical behaviour or fractal dimension of the branched polymer four different critical behaviours are found corresponding to random animal collapse or theta point percolation and compact cluster the theta point behaviour is described by a tricritical point while percolation corresponds to a higher order critical point where the effect of the solvent on the branched polymer is the same as the screening effect of the other clusters in percolation

lattice animal contact models of a collapsing branched polymer the authors consider two lattice animal models for the collapse of dilute branched polymers in a good solvent in both cases the collapse is driven by a near neighbour contact fugacity the two models differing in the way molecular weights are assessed either by the site content or the bond content of the animal they describe some rigorous results including bounds for the temperature dependence of their reduced limiting free energies on a d dimensional hypercubic lattice and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature of both models on a variety of lattices in addition they estimate the cross over exponent phi and find that for both models phi 0 60 or 0 03 d 2 and phi 0 82 or 0 02 d 3

phase diagram of branched polymer collapse the phase diagram of the collapse of a two dimensional infinite branched polymer interacting with the solvent and with itself through contact interactions is studied from the q ensuremath rightarrow 1 limit of an extension of the q state potts model exact solution on the bethe lattice and migdal kadanoff renormalization group calculations shows that there is a line of ensuremath theta transitions from the extended to a single compact phase the ensuremath theta line governed by three different fixed points consists of two lines of extended compact transitions which are in different universality classes and meet in a multicritical point on the other hand directed branched polymers are shown to be completely determined by the strongly embedded case and there is a single ensuremath theta transition which is in the directed percolation universality class textcopyright 1996 the american physical society

the cluster structure in collapsing animals in this paper i revisit the connection between edge percolation and the collapse transition in lattice animals it was shown by domb 1976 j phys a math gen 9 l141 that the critical percolation point is a transition in a certain model of self interacting animals i extend this result by showing that the free energy of lattice animals in the cycle contact model is non analytic in contact and cycle activities at points other than the critical percolation point this correspondence between percolation and collapsing animals suggests that the collapse transition in this model may be related to percolation contact collapse in animals is then studied from a percolation of clusters of contacts perspective i first argue that there is a critical activity for percolation of clusters of contacts and then investigate this numerically

phase diagram of inhomogeneous percolation with a defect plane let mathbb l be the d dimensional hypercubic lattice and let mathbb l 0 be an s dimensional sublattice with 2 le s p c d and a surface supercritical regime for p sigma p we show that sigma p is a strictly decreasing function for p in 0 p c d with a jump discontinuity at p c d we extend the aizenman barsky differential inequalities for homogeneous percolation to the inhomogeneous model and use them to prove that the susceptibility is finite inside the subcritical phase we prove that the cluster size distribution decays exponentially in the subcritical phase and sub exponentially in the supercritical phases for a model of lattice animals with a defect plane the free energy is related to functions of the inhomogeneous percolation model and we show how the percolation transition implies a non analyticity in the free energy of the animal model finally we present simulation estimates of the critical curve sigma p

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

collapse transition of randomly branched polymers renormalized field theory we present a minimal dynamical model for randomly branched isotropic polymers and we study this model in the framework of renormalized field theory for the swollen phase we show that our model provides a route to understand the well established dimensional reduction results from a different angle for the collapse transition we uncover a hidden becchi rouet stora supersymmetry signaling the sole relevance of tree configurations we correct the long standing one loop results for the critical exponents and we push these results on to two loop order for the collapse transition we find a runaway of the renormalization group flow which lends credence to the possibility that this transition is a fluctuation induced first order transition our dynamical model allows us to calculate for the first time the fractal dimension of the shortest path on randomly branched polymers in the swollen phase as well as at the collapse transition and related fractal dimensions

collapse transition and asymptotic scaling behavior of lattice animals low temperature expansion a low temperature expansion for the free energy density of lattice animals is derived analysis of the series yields a collapse transition temperature oftc 0 54 in close agreement with previous estimates it is demonstrated that p k the number ofp particle p bond animals obeys the asymptotic scaling law log p k pg k p o p the low temperature series and numerical data are used to estimate the scaling function

statistics of collapsed lattice animals rigorous results and monte carlo simulations in this paper a two dimensional model of collapsed lattice animals with nearest neighbour contact interactions is introduced and the limiting free energy for the model is proved to exist we also establish that the partition function for collapsed lattice animals is related to the reliability polynomial for the m m grid graph and to the generating function for spanning forests on the m m grid graph from these relationships and using results about directed animals we are able to obtain rigorous upper and lower bounds on the free energy for collapsed lattice animals in addition we present monte carlo evidence that the heat capacity associated with this model remains bounded in the thermodynamic limit in fact we do not find any evidence for a second order phase transition for collapsed lattice animals

polygonal polyominoes on the square lattice we study a proper subset of polyominoes called polygonal polyominoes which are defined to be self avoiding polygons containing any number of holes each of which is a self avoiding polygon the staircase polygon subset with staircase holes is also discussed the internal holes have no common vertices with each other nor any common vertices with the surrounding polygon there are no holes within holes we use the finite lattice method to count the number of polygonal polyominoes on the square lattice series have been derived for both the perimeter and area generating functions it is known that while the critical point is unchanged by a finite number of holes when the number of holes is unrestricted the critical point changes the area generating function coefficients grow exponentially with a growth constant greater than that for polygons with a finite number of holes but less than that of polyominoes we provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes also we prove that enumerating by perimeter the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds

collapse transitions in animals and vesicles we discuss the relationship between the solvent model of collapse in strongly and weakly embeddable lattice animals and a model recently introduced by vanderzande 1993 and by stella et al 1992 we show that there are important qualitative differences in the behaviour predicted by these two models in the compact regime

on the effective potential of duru kleinert path integrals we propose a new method to evaluate the effective potential in the path integral for the fixed energy amplitude as well as for the pseudotime evolution kernel in the formalism by duru and kleinert restriction to the postpoint or the prepoint prescriptions in formulating time sliced path integrals is avoided by leaving off the use of expectation values for correction terms this enables us to consider an arbitrary ordering prescription and to examine the ordering dependence of the effective potential to investigate parameter dependences we introduce the ordering parameter in addition to the splitting parameter in the formulation of the time sliced path integral the resulting path integrals are found to be independent of the ordering parameter although the explicit dependence given by a contribution proportional to 1 2 2 on the splitting parameter remains as an application we check the relationship between path integrals for the radial oscillator and the radial coulomb system in arbitrary dime

simulations of lattice animals and trees the scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions using as microscopic models lattice animals and lattice trees on simple hypercubic lattices as a stochastic sampling method we use a biased sequential sampling algorithm with re sampling similar to the pruned enriched rosenbluth method perm used extensively for linear polymers essentially we start simulating percolation clusters either site or bond re weigh them according to the animal tree ensemble and prune or branch the further growth according to a heuristic fitness function in contrast to previous applications of perm this fitness function is not the weight with which the actual configuration would contribute to the partition sum but is closely related to it we obtain high statistics of animals with up to several thousand sites in all dimension 2 d 9 in addition to the partition sum number of different animals we estimate gyration radii and numbers of perimeter sites in all dimensions we verify the parisi sourlas prediction and we verify all exactly known critical exponents in dimensions 2 3 4 and 8 in addition we present the hitherto most precise estimates for growth constants in d 3 for clusters with one site attached to an attractive surface we verify for d 3 the superuniversality of the cross over exponent at the adsorption transition predicted by janssen and lyssy but not for d 2 there we find 0 480 4 instead of the conjectured 1 2 finally we discuss the collapse of animals and trees arguing that our present version of the algorithm is also efficient for some of the models studied in this context but showing that it is not very efficient for the classical model for collapsing animals