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An efficient algorithm is presented to simulate the O(N) loop model on the square lattice for arbitrary values of $N>0$. The scheme combines the worm algorithm with a new data structure to resolve both the problem of loop crossings and the necessity of counting the number of loops at each Monte Carlo update. With the use of this scheme, the line of critical points (and other properties) of the O($N$) model on the square lattice for $0

Various two-dimensional O(N) models coupled to Euclidean quantum gravity, whose intrinsic dimension is four, are shown to belong to universality classes of nongravitating statistical models in a lower number of dimensions. It is speculated that the matching critical behaviors in the gravitating and dimensionally reduced models may be manifestations of the holographic principle., Comment: 12 pages, 1 figure

The loop gas approach to lattice field theory provides an alternative, geometrical description in terms of fluctuating loops. Statistical ensembles of random loops can be efficiently generated by Monte Carlo simulations using the worm update algorithm. In this paper, concepts from percolation theory and the theory of self-avoiding random walks are used to describe estimators of physical observables that utilize the nature of the worm algorithm. The fractal structure of the random loops as well as their scaling properties are studied. To support this approach, the O(1) loop model, or high-temperature series expansion of the Ising model, is simulated on a honeycomb lattice, with its known exact results providing valuable benchmarks., Comment: 34 pages, 12 figures; v2: 2 figures and 1 table added; v3: typo's corrected

The fractal structure of high-temperature graphs of the three-dimensional Ising and XY models is investigated by simulating these graphs directly on a cubic lattice and analyzing them with the help of percolation observables. The Ising graphs are shown to percolate right at the Curie critical point. The diverging length scale relevant to the graphs in the vicinity of the percolation threshold is shown to be provided by the spin correlation length. The fractal dimension of the high-temperature graphs at criticality is estimated to be $D = 1.7349(65)$ for the Ising and $D = 1.7626(66)$ for the XY model., Comment: 19 pages

These notes give examples of how suitably defined geometrical objects encode in their fractal structure thermal critical behavior. The emphasis is on the two-dimensional Potts model for which two types of spin clusters can be defined. Whereas the Fortuin-Kasteleyn clusters describe the standard critical behavior, the geometrical clusters describe the tricritical behavior that arises when including vacant sites in the pure Potts model. Other phase transitions that allow for a geometrical description discussed in these notes include the superfluid phase transition and Bose-Einstein condensation., Comment: 10 pages,9 figures. Contribution to the IV Brazilian Meeting on Simulational Physics (Ouro Preto, Minas Gerais, Brazil, 9 - 12 August 2005)

In these notes, the application of Feynman's sum-over-paths approach to thermal phase transitions is discussed. The paradigm of such a spacetime approach to critical phenomena is provided by the high-temperature expansion of spin models. This expansion, known as the hopping expansion in the context of lattice field theory, yields a geometric description of the phase transition in these models, with the thermal critical exponents being determined by the fractal structure of the high-temperature graphs. The graphs percolate at the thermal critical point and can be studied using purely geometrical observables known from percolation theory. Besides the phase transition in spin models and in the closely related $\phi^4$ theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory., Comment: 59 pages, 18 figures. Write-up of Ising Lectures presented at the National Academy of Sciences, Lviv, Ukraine, 2004. 2nd version: corrected typos

The relation between critical exponents, characterizing a continuous phase transition, and the fractal structure of physical lines, proliferating at the critical point, is established by considering the two-dimensional O($N$) spin model for which many exact results are available., Comment: 3 pages, 1 figure

The fractal structure and critical properties of the high-temperature graphs of the two-dimensional O($N)$ model close to criticality are investigated. Based on Monte Carlo simulations, De Gennes' results for polymer chains, corresponding to the limit $N \to 0$, are generalized to random loops for arbitrary $-2 \leq N \leq 2$. The loops are also studied close to their tricritical point, known as the $\Theta$ point in the context of polymers, where they collapse. The corresponding fractal dimensions are argued to be in one-to-one correspondence with those at the critical point, leading to an analytic prediction for the magnetic scaling dimension at the O($N)$ tricritical point., Comment: 4 pages, no figures; 2nd version: Introduction rewritten, comparison of prediction with recent high-precision Monte Carlo data with figure included, references added

In a Ginzburg-Landau theory with $n$ fields, the anomalous dimension of the gauge-invariant nonlocal order parameter defined by the long-distance limit of Dirac's gauge-invariant two-point function is calculated. The result is exact for all $n$ to first order in $\epsilon \equiv 4-d$, and for all $d\in (2,4)$ to first order in $1/n$, and coincides with the previously calculated gauge-dependent exponent in the Landau gauge., Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper http://www.physik.fu-berlin.de/~kleinert/356

The fractal structure of spin clusters and their boundaries in the critical two-dimensional (2D) Ising model is investigated numerically. The fractal dimensions of these geometrical objects are estimated by means of Monte Carlo simulations on relatively small lattices through standard finite-size scaling. The obtained results are in excellent agreement with theoretical predictions and partly provide significant improvements in precision over existing numerical estimates., Comment: 8 pages, 8 figures; v2: minor changes in text, various plots are put in one figure

The geometrical approach to phase transitions is illustrated by simulating the high-temperature representation of the Ising model on a square lattice., Comment: 5 pages, 3 figures, talk presented at Conference on Computational Physics 2004, Genoa, 1-4 September 2004; 2nd version: slightly expanded version

The tricritical behavior of the two-dimensional $q$-state Potts model with vacancies for $1\leq q \leq4$ is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The close connection between the critical properties of the pure model and the tricritical properties of the diluted model is shown to be reflected in an intimate relation between Fortuin-Kasteleyn and geometrical clusters: The same transformation mapping the two critical regimes onto each other also maps the two cluster types onto each other. The map conserves the central charge, so that both cluster types are in the same universality class. The geometrical picture is supported by a Monte Carlo simulation of the high-temperature representation of the Ising model ($q=2$). In this new numerical approach, closed graph configurations are generated by means of a Metropolis update algorithm, involving single plaquettes., Comment: 23 pages, 11 figures, 2nd version: references added, introduction partly rewritten, error estimates improved