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Register a new userScaling exponents for a monkey on a tree - fractal dimensions of randomly branched polymers
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We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to 2-loop order and, where available, compare them to numerical results.
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A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents $ u$ and $eta$, the crossover exponent $phi$, as well as the (related) wave-vector exponent $beta_q$, and the correction-to-scaling exponent $omega$ to second order in $epsilon_8=8-d$. These are compared with the authors recent $epsilon$-expansion results [{it Phys. Rev. B} {bf 62} (2000) 12338; {it Nucl. Phys. B} {bf 612} (2001) 340] for the general case of an $m$-axial Lifshitz point. It is shown that the expansions obtained here by a direct calculation for the isotropic ($m=d$) Lifshitz point all follow from the latter upon setting $m=8-epsilon_8$. This is so despite recent claims to the contrary by de Albuquerque and Leite [{it J. Phys. A} {bf 35} (2002) 1807].
In this work a new strategy is proposed in order to build analytic and microscopic models of fluctuating polymer rings subjected to topological constraints. The topological invariants used to fix these constraints belong to a wide class of the so-called numerical topological invariants. For each invariant it is possible to derive a field theory that describes the statistical behavior of knotted and linked polymer rings following a straightforward algorithm. The treatment is not limited to the partition function of the system, but it allows also to express the expectation values of general observables as field theory amplitudes. Our strategy is illustrated taking as examples the Gauss linking number and a topological invariant belonging to a class of invariants due to Massey. The consistency of the new method developed here is checked by reproducing a previous field theoretical model of two linked polymer rings. After the passage to field theory, the original topological constraints imposed on the fluctuating paths of the polymers become constraints over the configurations of the topological fields that mediate the interactions of topological origin between the monomers. These constraints involve quantities like the cross-helicity which are of interest in other disciplines, like for instance in modeling the solar magnetic field. While the calculation of the vacuum expectation values of generic observables remains still challenging due to the complexity of the problem of topological entanglement in polymer systems, we succeed here to reduce the evaluation of the moments of the Gauss linking number for two linked polymer rings to the computation of the amplitudes of a free field theory.
We compute the crossover exponents of all quadratic and cubic deformations of critical field theories with permutation symmetry $S_q$ in $d=6-epsilon$ (Landau-Potts field theories) and $d=4-epsilon$ (hypertetrahedral models) up to three loops.We use our results to determine the $epsilon$-expansion of the fractal dimension of critical clusters in the most interesting cases, which include spanning trees and forests ($qto0$), and bond percolations ($qto1$). We also explicitly verify several expected degeneracies in the spectrum of relevant operators for natural values of $q$ upon analytic continuation, which are linked to logarithmic corrections of CFT correlators, and use the $epsilon$-expansion to determine the universal coefficients of such logarithms.
The exact solution of directed self-avoiding walks confined to a slit of finite width and interacting with the walls of the slit via an attractive potential has been calculated recently. The walks can be considered to model the polymer-induced steric stabilisation and sensitised floculation of colloidal dispersions. The large width asymptotics led to a phase diagram different to that of a polymer attached to, and attracted to, a single wall. The question that arises is: can one interpolate between the single wall and two wall cases? In this paper we calculate the exact scaling functions for the partition function by considering the two variable asymptotics of the partition function for simultaneous large length and large width. Consequently, we find the scaling functions for the force induced by the polymer on the walls. We find that these scaling functions are given by elliptic theta-functions. In some parts of the phase diagram there is more a complex crossover between the single wall and two wall cases and we elucidate how this happens.
We analyze the phase diagrams of self-avoiding walk models of uniform branched polymers adsorbed at a surface and subject to an externally applied vertical pulling force which, at critical values, desorbs the polymer. In particular, models of adsorbed branched polymers with homeomorphism types stars, tadpoles, dumbbells and combs are examined. These models generalize earlier results on linear, ring and $3$-star polymers. In the case of star polymers we confirm a phase diagram with four phases (a free, an adsorbed, a ballistic, and a mixed phase) first seen in the paper by Janse van Rensburg EJ and Whittington SG 2018 J. Phys. A: Math. Theor. 51 204001 for $3$-star polymers. The phase diagram of tadpoles may include four phases (including a mixed phase) if the tadpole is pulled from the adsorbing surface by the end vertex of its tail. If it is instead pulled from the middle vertex of its head, then there are only three phases (the mixed phase is absent). For a dumbbell pulled from the middle vertex of a ring, there are only three phases. For combs with $t$ teeth there are four phases, independent of the value of $t$ for all $t ge 1$.
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