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Petter Minnhagen

Publications and source records attributed to Petter Minnhagen.

At least 37 records · Page 2Linked to original sources
arXiv2010

Analytic results for the percolation transitions of the enhanced binary tree

Percolation for a planar lattice has a single percolation threshold, whereas percolation for a negatively curved lattice displays two separate thresholds. The enhanced binary tree (EBT) can be viewed as a prototype model displaying two separate percolation thresholds. We present an analytic result for the EBT model which gives two critical percolation threshold probabilities, $p_{c1}=1/2\sqrt{13}-3/2$ and $p_{c2}=1/2$, and yields a size-scaling exponent $Φ=\ln [\frac{p(1+p)}{1-p(1-p)}]/\ln 2$. It is inferred that the two threshold values give exact upper limits and that $p_{c1}$ is furthermore exact. In addition, we argue that $p_{c2}$ is also exact. The physics of the model and the results are described within the midpoint-percolation concept: Monte Carlo simulations are presented for the number of boundary points which are reached from the midpoint, and the results are compared to the number of routes from the midpoint to the boundary given by the analytic solution. These comparisons provide a more precise physical picture of what happens at the transitions. Finally, the results are compared to related works, in particular, the Cayley tree and Monte Carlo results for hyperbolic lattices as well as earlier results for the EBT model. It disproves a conjecture that the EBT has an exact relation to the thresholds of its dual lattice.

cond-mat.stat-mech↗
arXiv2010

Non-Kosterlitz-Thouless transitions for the $q$-state clock models

The $q$-state clock model with the cosine potential has a single phase transition for $q\leq4$ and two transitions for $q\geq5$. It is shown by Monte Carlo simulations that the helicity modulus for the five-state clock model ($q=5$) does not vanish at the high-temperature transition. This is in contrast to the clock models with $q\geq6$ for which the helicity modulus vanishes. This means that the transition for the five-state clock model differs from the Kosterlitz-Thouless (KT) transition. It is also shown that this change in the transition is caused by an interplay between the number of angular directions and the interaction potential: by slightly modifying the interaction potential, the KT transition for $q=6$ turns into the same non-KT transition. Likewise, the KT transition is recovered for $q=5$ when the Villain potential is used. Comparisons with other clock-model results are made and discussed.

cond-mat.stat-mech↗
arXiv2010

Surface and bulk criticality in midpoint percolation

The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular $d$-dimensional hypercubic lattices are in the present work investigated using the same concept. Specifically, the site-percolation transitions at the critical thresholds are investigated for dimensions up to $d=10$ by means of the Leath algorithm. It is shown that the explicit inclusion of the boundaries provides a straightforward way to obtain critical indices, both for the bulk and surface parts. At and above the critical dimension $d=6$, it is found that the percolation cluster contains only a finite number of surface points in the infinite-size limit. This is in accordance with the expectation from studies of lattices with negative curvature. It is also found that the number of surface points, reached by the percolation cluster in the infinite limit, approaches 2d for large dimensions $d$. We also note that the size dependence in proliferation of percolating clusters for $d\ge 7$ can be obtained by solely counting surface points of the midpoint cluster.

cond-mat.stat-mech↗
arXiv2009

True and quasi long-range order in the generalized $q$-state clock model

From consideration of the order-parameter distribution, we propose an observable which makes a clear distinction between true and quasi long-range orders in the two-dimensional generalized $q$-state clock model. Measuring this quantity by Monte Carlo simulations for $q=8$, we construct a phase diagram and identify critical properties across the phase-separation lines among the true long-range order, quasi long-range order, and disorder. Our result supports the theoretical prediction that there appears a discontinuous order-disorder transition as soon as the two phase-separation lines merge.

cond-mat.stat-mech↗
arXiv2009

Comment on `Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees'

The enhanced binary tree (EBT) is a nontransitive graph which has two percolation thresholds $p_{c1}$ and $p_{c2}$ with $p_{c1}<p_{c2}$. Our Monte Carlo study implies that the second threshold $p_{c2}$ is significantly lower than a recent claim by Nogawa and Hasegawa (J. Phys. A: Math. Theor. {\bf 42} (2009) 145001). This means that $p_{c2}$ for the EBT does not obey the duality relation for the thresholds of dual graphs $p_{c2}+\overline{p}_{c1}=1$ which is a property of a transitive, nonamenable, planar graph with one end. As in regular hyperbolic lattices, this relation instead becomes an inequality $p_{c2}+\overline{p}_{c1}<1$. We also find that the critical behavior is well described by the scaling form previously found for regular hyperbolic lattices.

cond-mat.stat-mech↗
arXiv2009

The meta book and size-dependent properties of written language

Evidence is given for a systematic text-length dependence of the power-law index gamma of a single book. The estimated gamma values are consistent with a monotonic decrease from 2 to 1 with increasing length of a text. A direct connection to an extended Heap's law is explored. The infinite book limit is, as a consequence, proposed to be given by gamma = 1 instead of the value gamma=2 expected if the Zipf's law was ubiquitously applicable. In addition we explore the idea that the systematic text-length dependence can be described by a meta book concept, which is an abstract representation reflecting the word-frequency structure of a text. According to this concept the word-frequency distribution of a text, with a certain length written by a single author, has the same characteristics as a text of the same length pulled out from an imaginary complete infinite corpus written by the same author.

physics.soc-ph↗
arXiv2009

Phase transition of $q$-state clock models on heptagonal lattices

We study the $q$-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every $q \ge 2$. The persistence of the third phase for all $q$ is in contrast with the disappearance of the counterpart phase in a planar system for small $q$, which indicates the significance of nonvanishing surface-volume ratio that is peculiar in the heptagonal lattice. Analytic arguments based on Ginzburg-Landau theory and generalized Cayley trees make clear that the two-stage transition in the present system is attributed to an energy gap of spin-wave excitations and strong boundary-spin contributions. We further demonstrate that boundary effects breaks the mean-field character in the bulk region, which establishes the consistency with results of clock models on boundary-free hyperbolic lattices.

cond-mat.stat-mech↗
arXiv2009

Flow improvement caused by agents who ignore traffic rules

A system of agents moving along a road in both directions is studied numerically within a cellular-automata formulation. An agent steps to the right with probability $q$ or to the left with $1-q$ when encountering other agents. Our model is restricted to two agent types, traffic-rule abiders ($q=1$) and traffic-rule ignorers ($q=1/2$). The traffic flow, resulting from the interaction between these two types of agents, is obtained as a function of density and relative fraction. The risk for jamming at a fixed density, when starting from a disordered situation, is smaller when every agent abides by a traffic rule than when all agents ignore the rule. Nevertheless, the absolute minimum occurs when a small fraction of ignorers are present within a majority of abiders. The characteristic features for the spatial structure of the flow pattern are obtained and discussed.

physics.soc-ph↗
arXiv2009

Size dependent word frequencies and translational invariance of books

It is shown that a real novel shares many characteristic features with a null model in which the words are randomly distributed throughout the text. Such a common feature is a certain translational invariance of the text. Another is that the functional form of the word-frequency distribution of a novel depends on the length of the text in the same way as the null model. This means that an approximate power-law tail ascribed to the data will have an exponent which changes with the size of the text-section which is analyzed. A further consequence is that a novel cannot be described by text-evolution models like the Simon model. The size-transformation of a novel is found to be well described by a specific Random Book Transformation. This size transformation in addition enables a more precise determination of the functional form of the word-frequency distribution. The implications of the results are discussed.

cs.CL↗
arXiv2009

Multiscaling in the YX model of networks

We investigate a Hamiltonian model of networks. The model is a mirror formulation of the XY model (hence the name) -- instead letting the XY spins vary, keeping the coupling topology static, we keep the spins conserved and sample different underlying networks. Our numerical simulations show complex scaling behaviors, but no finite-temperature critical behavior. The ground state and low-order excitations for sparse, finite graphs is a fragmented set of isolated network clusters. Configurations of higher energy are typically more connected. The connected networks of lowest energy are stretched out giving the network large average distances. For the finite sizes we investigate there are three regions -- a low-energy regime of fragmented networks, and intermediate regime of stretched-out networks, and a high-energy regime of compact, disordered topologies. Scaling up the system size, the borders between these regimes approach zero temperature algebraically, but different network structural quantities approach their T=0-values with different exponents.

physics.comp-ph↗
arXiv2009

Percolation on hyperbolic lattices

The percolation transitions on hyperbolic lattices are investigated numerically using finite-size scaling methods. The existence of two distinct percolation thresholds is verified. At the lower threshold, an unbounded cluster appears and reaches from the middle to the boundary. This transition is of the same type and has the same finite-size scaling properties as the corresponding transition for the Cayley tree. At the upper threshold, on the other hand, a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections. The finite-size scaling properties for this upper threshold are different from those of the Cayley tree and two of the critical exponents are obtained. The results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own.

cond-mat.stat-mech↗
arXiv2008

Symmetry-allowed phase transitions realized by the two-dimensional fully frustrated XY class

A 2D Fully Frustrated XY(FFXY) class of models is shown to contain a new groundstate in addition to the checkerboard groundstates of the standard 2D FFXY model. The spin configuration of this additional groundstate is obtained. Associated with this groundstate there are additional phase transitions. An order parameter accounting for these new transitions is proposed. The transitions associated with the new order parameter are suggested to be similar to a 2D liquid-gas transition which implies Z_2-Ising like transitions. This suggests that the class of 2D FFXY models belongs within a U(1) x Z_2 x Z_2-designation of possible transitions, which implies that there are seven different possible single and combined transitions. MC-simulations for the generalized fully frustrated XY (GFFXY) model on a square lattice are used to investigate which of these possibilities can be realized in practice: five of the seven are encountered. Four critical points are deduced from the MC-simulations, three consistent with central charge c=3/2 and one with c=1. The implications for the standard 2D FFXY-model are discussed in particular with respect to the long standing controversy concerning the characteristics of its phase transitions.

cond-mat.stat-mech↗
arXiv2008

The Blind Watchmaker Network: Scale-freeness and Evolution

It is suggested that the degree distribution for networks of the cell-metabolism for simple organisms reflects an ubiquitous randomness. This implies that natural selection has exerted no or very little pressure on the network degree distribution during evolution. The corresponding random network, here termed the blind watchmaker network has a power-law degree distribution with an exponent gamma >= 2. It is random with respect to a complete set of network states characterized by a description of which links are attached to a node as well as a time-ordering of these links. No a priory assumption of any growth mechanism or evolution process is made. It is found that the degree distribution of the blind watchmaker network agrees very precisely with that of the metabolic networks. This implies that the evolutionary pathway of the cell-metabolism, when projected onto a metabolic network representation, has remained statistically random with respect to a complete set of network states. This suggests that even a biological system, which due to natural selection has developed an enormous specificity like the cellular metabolism, nevertheless can, at the same time, display well defined characteristics emanating from the ubiquitous inherent random element of Darwinian evolution. The fact that also completely random networks may have scale-free node distributions gives a new perspective on the origin of scale-free networks in general.

physics.bio-ph↗
arXiv2008

Phase diagram of generalized fully frustrated XY model in two dimensions

It is shown that the phase diagram of the two-dimensional generalized fully-frustrated XY model on a square lattice contains a crossing of the chirality transition and the Kosterlitz-Thouless (KT) transition, as well as a stable phase characterized by a finite helicity modulus $Υ$ and an unbroken chirality symmetry. The crossing point itself is consistent with a critical point without any jump in $Υ$, with the size ($L$) scaling $% Υ\sim L^{-0.63}$ and the critical index $ν\approx0.77$. The KT transition line remains continuous beyond the crossing but eventually turns into a first-order line. The results are established using Monte-Carlo simulations of the staggered magnetization, helicity modulus, and the fourth-order helicity modulus.

cond-mat.supr-con↗
arXiv2007

Optimization and Scale-freeness for Complex Networks

Complex networks are mapped to a model of boxes and balls where the balls are distinguishable. It is shown that the scale-free size distribution of boxes maximizes the information associated with the boxes provided configurations including boxes containing a finite fraction of the total amount of balls are excluded. It is conjectured that for a connected network with only links between different nodes, the nodes with a finite fraction of links are effectively suppressed. It is hence suggested that for such networks the scale-free node-size distribution maximizes the information encoded on the nodes. The noise associated with the size distributions is also obtained from a maximum entropy principle. Finally explicit predictions from our least bias approach are found to be born out by metabolic networks.

cond-mat.stat-mech↗
arXiv2007

Dynamic scaling regimes of collective decision making

We investigate a social system of agents faced with a binary choice. We assume there is a correct, or beneficial, outcome of this choice. Furthermore, we assume agents are influenced by others in making their decision, and that the agents can obtain information that may guide them towards making a correct decision. The dynamic model we propose is of nonequilibrium type, converging to a final decision. We run it on random graphs and scale-free networks. On random graphs, we find two distinct regions in terms of the "finalizing time" -- the time until all agents have finalized their decisions. On scale-free networks on the other hand, there does not seem to be any such distinct scaling regions.

q-bio.PE↗
arXiv2007

Phase Transition of XY Model in Heptagonal Lattice

We numerically investigate the nature of the phase transition of the XY model in the heptagonal lattice with the negative curvature, in comparison to other interaction structures such as a flat two-dimensional (2D) square lattice and a small-world network. Although the heptagonal lattice has a very short characteristic path length like the small-world network structure, it is revealed via calculation of the Binder's cumulant that the former exhibits a zero-temperature phase transition while the latter has the finite-temperature transition of the mean-field nature. Through the computation of the vortex density as well as the correlation function in the low-temperature approximation, we show that the absence of the phase transition originates from the strong spinwave-type fluctuation, which is discussed in relation to the usual 2D XY model.

cond-mat.stat-mech↗