The Hausdorff dimension of the Julia sets concerning generated renormalization transformation

<abstract><p>Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $\end{document} </tex-math></disp-formula></p> <p>where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.</p></abstract>

Some remarks on the geometry of circle maps with a break point

We study circle homeomorphisms f ? C2(S1\{xb}) whose rotation number ?f is irrational, with a single break point xb at which f' has a jump discontinuity. We prove that the behavior of the ratios of the lengths of any two adjacent intervals of the dynamical partition depends on the size of break and on the continued fraction decomposition of ?f. We also prove a result analogous to Yoccoz?s lemma on the asymptotic behaviour of the lengths of the intervals of trajectories of the renormalization transformation Rn(f).

The Lognormal Distribution and Quantum Monte Carlo Data

Quantum Monte Carlo data are often afflicted with distributions that resemble lognormal probability distributions and consequently their statistical analysis cannot be based on simple Gaussian assumptions. To this extent a method is introduced to estimate these distributions and thus give better estimates to errors associated with them. This method entails reconstructing the probability distribution of a set of data, with given mean and variance, that has been assumed to be lognormal prior to undergoing a blocking or renormalization transformation. In doing so, we perform a numerical evaluation of the renormalized sum of lognormal random variables. This technique is applied to a simple quantum model utilizing the single-thread Monte Carlo algorithm to estimate the ground state energy or dominant eigenvalue of a Hamiltonian matrix.

DOMAIN WALL RENORMALIZATION GROUP ANALYSIS OF TWO-DIMENSIONAL ISING MODEL

Using a recently proposed new renormalization group method (tensor renormalization group), we analyze the Ising model on the two-dimensional square lattice. For the lowest-order approximation with two-domain wall states, it realizes the idea of coarse graining of domain walls. We write down explicit analytic renormalization transformation and prove that the picture of the coarse graining of the physical domain walls does hold for all physical renormalization group flows. We solve it to get the fixed point structure and obtain the critical exponents and the critical temperature. These results are very near to the exact values. We also briefly report the improvement using four-domain wall states.