arXiv subjects
Publications and source records attributed to K. Pinn.
I explicitly work out closed form solutions for the optimal hedging strategies (in the sense of Bouchaud and Sornette) in the case of European call options, where the underlying is modeled by (unbiased) iid additive returns with Student-t distributions. The results may serve as illustrative examples for option pricing in the presence of fat tails.
I present a calculation of the interfacial width within the capillary wave (Gaussian) approximation. The calculation is done on rectangular lattices of size L_1 times L_2, with periodic boundary conditions.
I study the recurrence D(n)= D(D(n-1))+D(n-1-D(n-2)), D(1)=D(2)=1. Its definition has some similarity to that of Conway's sequence defined through a(n)= a(a(n-1))+a(n-a(n-1)), a(1)=a(2)=1. However, in contradistinction to the completely regular and predictable behaviour of a(n), the D-numbers exhibit chaotic patterns. In its statistical properties, the D-sequence shows striking similarities with Hofstadter's Q(n)-sequence, defined through Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2)), Q(1)=Q(2)=1. Compared to the Hofstadter sequence, the D-recurrence shows higher structural order. It is organized in well-defined ``generations'', separated by smooth and predictable regions. The article is complemented by a study of two further recurrence relations with definitions similar to those of the Q-numbers. There is some evidence that the different sequences studied share a universality class. Could it be that there are some real life processes modelled by these recurrences? I OFFER A CASH PRIZE OF $100 TO THE FIRST PROVIDING A PROOF OF SOME CONJECTURES ABOUT D(n) FORMULATED IN THIS ARTICLE.
Effective Boltzmannians in the sense of the block spin renormalization group are computed for the 2D Ising model. The blocking is done with majority and Kadanoff rules for blocks of size 2 by 2. Transfer matrix techniques allow the determination of the effective Boltzmannians as polynomials in u=exp(4 Beta) for lattices of up to 4 by 4 blocks. The zeroes of these polynomials are computed for all non-equivalent block spin configurations. Their distribution in the complex Beta plane reflects the regularity structure of the block spin transformation. In the case of the Kadanoff rule spurious zeroes approach the positive real Beta axis at large values of Beta. They might be related to the renormalization group pathologies discussed in the literature.
We investigate local update algorithms for the fully frustrated XY model on a square lattice. In addition to the standard updating procedures like the Metropolis or heat bath algorithm we include overrelaxation sweeps, implemented through single spin updates that preserve the energy of the configuration. The dynamical critical exponent (of order two) stays more or less unchanged. However, the integrated autocorrelation times of the algorithm can be significantly reduced.
A number of observations are made on Hofstadter's integer sequence defined by Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2)), for n > 2, and Q(1)=Q(2)=1. On short scales the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k-th generation has 2**k members which have ``parents'' mostly in generation k-1, and a few from generation k-2. In this sense the series becomes Fibonacci type on a logarithmic scale. The mean square size of S(n)=Q(n)-n/2, averaged over generations is like 2**(alpha*k), with exponent alpha = 0.88(1). The probability distribution p^*(x) of x = R(n)= S(n)/n**alpha, n >> 1, is well defined and is strongly non-Gaussian. The probability distribution of x_m = R(n)-R(n-m) is given by p_m(x_m)= lambda_m * p^*(x_m/lambda_m). It is conjectured that lambda_m goes to sqrt(2) for large m.
We propose a method to obtain an improved Hamiltonian (action) for the Ising universality class in three dimensions. The improved Hamiltonian has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the +1,-1 Ising model with nearest neighbour and body diagonal interaction, the spin-1 model with states 0,+1,-1, and nearest neighbour interaction, and phi**4-theory on the lattice (Landau-Ginzburg Hamiltonian). The remarkable finite size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are nu= 0.6298(5) and eta= 0.0366(8), where the given error estimates take into account the statistical and systematic uncertainties.
We compute renormalization group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing non-trivial fixed points for a three-dimensional real $N$-component field: the O(N)-invariant fixed point vs.~the cubic-invariant fixed point. We compute the critical value $N_{c}$ of the cubic $\phi^{4}$-perturbation at the O(N)-fixed point. The O(N) fixed point is stable under a cubic $\phi^{4}$-perturbation below $N_{c}$, above $N_{c}$ it is unstable. The critical value comes out as $2.219435<N_{c}< 2.219436$ in the ultralocal approximation. We also compute the critical value of $N$ at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.
We present results from the simulation of a two-coupling spin-1 model with states 0,+1,-1 and nearest neighbour interaction. By a suitable choice of couplings we are able to drastically reduce the effects of corrections to scaling. Our estimates for the critical exponents are nu= 0.6299(3) and eta = 0.0359(10). For the Binder cumulant related universal ratio we obtain Q= 0.6240(2). The universal ratio of partition functions with antiperiodic/periodic boundary conditions, respectively, is Z_a/Z_p = 0.5425(2).
There are 880 magic squares of size 4 by 4, and 275,305,224 of size 5 by 5. It seems very difficult if not impossible to count exactly the number of higher order magic squares. We propose a method to estimate these numbers by Monte Carlo simulating magic squares at finite temperature. One is led to perform low temperature simulations of a system with many ground states that are separated by energy barriers. The Parallel Tempering Monte Carlo method turns out to be of great help here. Our estimate for the number of 6 by 6 magic squares is 0.17745(16) times 10**20.
The Svetitsky-Yaffe conjecture for finite temperature 4d SU(2) lattice gauge theory is confirmed by observing matching of block spin effective actions of the gauge model with those of the 3d Ising model. The effective action for the gauge model is defined by blocking the signs of the Polyakov loops with the majority rule. To compute it numerically, we apply a variant of the IMCRG method of Gupta and Cordery.
We present estimates for the 3D Ising model on the cubic lattice, both regarding interface and bulk properties. We have results for the interface tension, in particular the amplitude sigma0 in the critical law sigma = sigma0*t**mu, and for the universal combination R_ = sigma*xi**2. Concerning the bulk properties, we estimate the specific heat universal amplitude ratio A+/A_, together with the exponent alpha, the nonsingular background of energy and specific heat at criticality, together with the exponent nu. There are also results for the universal combination fs*xi**3, where fs is the singular part of the free energy.
Using a variant of the IMCRG method of Gupta and Cordery, we explicitly compute majority rule block spin effective actions for the signs of the Polyakov loops in 4D SU(2) finite temperature lattice gauge theories. To the best of our knowledge, this is the first attempt to compute numerically effective actions for the Polyakov loop degrees of freedom in 4D SU(2). The most important observations are: 1. The renormalization group flow at the deconfinement transition can be nicely matched with the flow of the 3D Ising model, thus confirming the Svetitsky-Yaffe conjecture. 2. The IMCRG simulations of the FT SU(2) model have strongly reduced critical slowing down.
We analyse Monte Carlo data for the energy and specific heat at and close to the critical point of the 3D cubic Ising model. From the finite size scaling of the energy E and the specific heat C at criticality we obtain the estimate nu = 0.6308(10). Furthermore, one obtains precise estimates for the ``backgrounds'' (nonsingular parts) E_ns and C_ns. Fitting solely off critical energy estimates to a scaling law, we find depending on the choice of the reduced temperature, either A_+/A_- = 0.550(12) and alpha=0.1115(37), or A_+/A_- = 0.567(16) and alpha=0.1047(48). Including information from the data at T_c, we obtain the estimate A_+/A_- = 0.560(10). We also determine the universal combination f_s xi^3 in both phases.
Using the Monte Carlo method, we determine the free energy of the interface of the 3D Ising model in the scaling region. By integrating the interface energies over the inverse temperature $\beta$, we obtain estimates for the free energies of interfaces with cross sections up to 96 by 96, and for a range $0.223 \leq \beta \leq 0.23$. Our data yield a precise estimation of the interface tensions $\sigma$. We determine the amplitude $\sigma_0$ in the critical law $\sigma \sim \sigma_0 t^{\mu}$ and estimate the combination $\sigma \xi^2$ which yields the universal constant $R_{-}$ in the critical limit.
We study the roughening transition of the dual of the 2D XY model, of the Discrete Gaussian model, of the Absolute Value Solid-On-Solid model and of the interface in an Ising model on a 3D simple cubic lattice. The investigation relies on a renormalization group finite size scaling method that was proposed and successfully tested a few years ago. The basic idea is to match the renormalization group flow of the interface observables with that of the exactly solvable BCSOS model. Our estimates for the critical couplings are $β_R^{XY}=1.1199(1)$, $K_R^{DG}=0.6653(2)$ and $K_R^{ASOS}=0.80608(2)$ for the XY-model, the Discrete Gaussian model and the Absolute Value Solid-On-Solid model, respectively. For the inverse roughening temperature of the Ising interface we find $K_R^{Ising}= 0.40758(1)$. To the best of our knowledge, these are the most precise estimates for these parameters published so far.
We consider the question of removing the ultraviolet cutoff in a 2D Quantum Field Theory with an interaction term which is non-renormalizable by power counting. This model arises as the first non-trivial correction beyond the Gaussian approximation of the so called Capillary Wave or Drumhead Model, and is rather important from a physical point of view since it correctly describes the finite size effects of two-dimensional interfaces. Despite the fact that the interaction is non-renormalizable, we prove that for a large class of regularization schemes the finite and divergent parts can be separated in a simple way. Furthermore, the finite part is independent of the choice of cutoff prescription used.
We study the iteration of block spin transformations in the O(3) symmetric non-linear sigma-model on a two-dimensional square lattice with help of the Monte Carlo method. In contrast to the classical Monte Carlo Renormalization Group approach, we do attempt to explicitly compute the block spin effective actions. Using two different methods for the determination of effective couplings, we study the renormalization group flow for various parametrization and truncation schemes. The largest ansatz for the effective action contains thirteen coupling constants. Actions on the renormalized trajectory should describe theories with no lattice artefacts, even at small correlation length. However, tests with the step scaling function of Luescher et al. reveal that our truncated effective actions show sizable scaling violations indicating that the ansaetze are still too small.