Ferromagnetic phase transition for the spanning-forest model (q \to 0 limit of the Potts model) in three or more dimensions

We present Monte Carlo simulations of the spanning-forest model (q \to 0 limit of the ferromagnetic Potts model) in spatial dimensions d=3,4,5. We show that, in contrast to the two-dimensional case, the model has a "ferromagnetic" second-order phase transition at a finite positive value w_c. We present numerical estimates of w_c and of the thermal and magnetic critical exponents. We conjecture that the upper critical dimension is 6.Comment: LaTex2e, 4 pages; includes 6 Postscript figures; Version 2 has expanded title as published in PR

Cluster simulations of loop models on two-dimensional lattices

We develop cluster algorithms for a broad class of loop models on two-dimensional lattices, including several standard O(n) loop models at n \ge 1. We show that our algorithm has little or no critical slowing-down when 1 \le n \le 2. We use this algorithm to investigate the honeycomb-lattice O(n) loop model, for which we determine several new critical exponents, and a square-lattice O(n) loop model, for which we obtain new information on the phase diagram.Comment: LaTex2e, 4 pages; includes 1 table and 2 figures. Totally rewritten in version 2, with new theory and new data. Version 3 as published in PR

Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z \ge \alpha/\nu is close to but probably not sharp in d=2, and is far from sharp in d=3, for all q. The conjecture z \ge \beta/\nu is false (for some values of q) in both d=2 and d=3.Comment: Revtex4, 4 pages including 4 figure

Grassmann Integral Representation for Spanning Hyperforests

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J. Phys.

Critical speeding-up in a local dynamics for the random-cluster model

We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2,3, by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S_2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep, so that the integrated autocorrelation time tends to zero as the critical point is approached. We also show that the dynamic critical exponent z_{exp} is very close (possibly equal) to the rigorous lower bound \alpha/\nu, and quite possibly smaller than the corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics.Comment: LaTex2e/revtex4, 4 pages, includes 5 figure

Equilibrium fluid-solid coexistence of hard spheres

We present a tethered Monte Carlo simulation of the crystallization of hard spheres. Our method boosts the traditional umbrella sampling to the point of making practical the study of constrained Gibb's free energies depending on several crystalline order-parameters. We obtain high-accuracy estimates of the fluid-crystal coexistence pressure for up to 2916 particles (enough to accommodate fluid-solid interfaces). We are able to extrapolate to infinite volume the coexistence pressure (p_{co}=11.5727(10) k_B T/\sigma^3) and the interfacial free energy (\gamma_{100}=0.636(11) k_B T/\sigma^2).Comment: 6 pages, 4 pdf figures. Version to be published in PRL. Appendices contain Supplemental Materia

Renormalised four-point coupling constant in the three-dimensional O(N) model with N=0

We simulate self-avoiding walks on a cubic lattice and determine the second virial coefficient for walks of different lengths. This allows us to determine the critical value of the renormalized four-point coupling constant in the three-dimensional N-vector universality class for N=0. We obtain g* = 1.4005(5), where g is normalized so that the three-dimensional field-theoretical beta-function behaves as \beta(g) = - g + g^2 for small g. As a byproduct, we also obtain precise estimates of the interpenetration ratio Psi*, Psi* = 0.24685(11), and of the exponent \nu, \nu = 0.5876(2).Comment: 16 page

Topological interactions between ring polymers: Implications for chromatin loops

Chromatin looping is a major epigenetic regulatory mechanism in higher eukaryotes. Besides its role in transcriptional regulation, chromatin loops have been proposed to play a pivotal role in the segregation of entire chromosomes. The detailed topological and entropic forces between loops still remain elusive. Here, we quantitatively determine the potential of mean force between the centers of mass of two ring polymers, i.e. loops. We find that the transition from a linear to a ring polymer induces a strong increase in the entropic repulsion between these two polymers. On top, topological interactions such as the non-catenation constraint further reduce the number of accessible conformations of close-by ring polymers by about 50%, resulting in an additional effective repulsion. Furthermore, the transition from linear to ring polymers displays changes in the conformational and structural properties of the system. In fact, ring polymers adopt a markedly more ordered and aligned state than linear ones. The forces and accompanying changes in shape and alignment between ring polymers suggest an important regulatory function of such a topology in biopolymers. We conjecture that dynamic loop formation in chromatin might act as a versatile control mechanism regulating and maintaining different local states of compaction and order.Comment: 12 pages, 11 figures. The article has been accepted by The Journal Of Chemical Physics. After it is published, it will be found at http://jcp.aip.or

Self-avoiding walks on scale-free networks

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are expected to be more suitable than unrestricted random walks to explore various kinds of real-life networks. Here we study long-range properties of random SAWs on scale-free networks, characterized by a degree distribution $P(k) \sim k^{-\gamma}$. In the limit of large networks (system size $N \to \infty$), the average number $s_n$ of SAWs starting from a generic site increases as $\mu^n$, with $\mu = / - 1$. For finite $N$, $s_n$ is reduced due to the presence of loops in the network, which causes the emergence of attrition of the paths. For kinetic growth walks, the average maximum length, $$, increases as a power of the system size: $\sim N^{\alpha}$, with an exponent $\alpha$ increasing as the parameter $\gamma$ is raised. We discuss the dependence of $\alpha$ on the minimum allowed degree in the network. A similar power-law dependence is found for the mean self-intersection length of non-reversal random walks. Simulation results support our approximate analytical calculations.Comment: 9 pages, 7 figure