arXiv subjects

P. M. Duxbury

Publications and source records attributed to P. M. Duxbury.

14 recordsLinked to original sources

arXiv2008

Dynamics of k-core percolation

In many network applications nodes are stable provided they have at least k neighbors, and a network of k-stable nodes is called a k-core. The vulnerability to random attack is characterized by the size of culling avalanches which occur after a randomly chosen k-core node is removed. Simulations of lattices in two, three and four dimensions, as well as small world networks, indicate that power-law avalanches occur in first order k-core systems, while truncated avalanches are characteristic of second order cases.

cond-mat.stat-mech↗

arXiv2003

Percolation of frozen order in glassy combinatorial problems

A local order parameter which is important in the analysis of phase transitions in frustrated combinatorial problems is the probability that a node is frozen in a particular state. There is a percolative transition when an infinite connected cluster of these frozen nodes emerges. In this contribution, we develop theories based on this percolation process and discuss its relation to conventional connectivity percolation and its generalisation to k-connectivity percolation. The emergence of frozen order may also be considered to be a form of constraint percolation (CP) which enables us to draw analogies with rigidity percolation and its associated matching problems. We show that very simple CP processes on Bethe lattices lead to the replica symmetric equations for KSAT, coloring and the Viana-Bray model.

cond-mat.dis-nn↗

arXiv2003

Precise polynomial heuristic for an NP-complete problem

We introduce a simple, efficient and precise polynomial heuristic for a key NP complete problem, minimum vertex cover. Our method is iterative and operates in probability space. Once a stable probability solution is found we find the true combinatorial solution from the probabilities. For system sizes which are amenable to exact solution by conventional means, we find a correct minimum vertex cover for all cases which we have tested, which include random graphs and diluted triangular lattices of up to 100 sites. We present precise data for minimum vertex cover on graphs of up to 50,000 sites. Extensions of the method to hard core lattices gases and other NP problems are discussed.

cond-mat.stat-mech↗

arXiv2002

Structural compliance, misfit strain and stripe nanostructures in cuprate superconductors

Structural compliance is the ability of a crystal structure to accommodate variations in local atomic bond-lengths without incurring large strain energies. We show that the structural compliance of cuprates is relatively small, so that short, highly doped, Cu-O-Cu bonds in stripes are subject to a tensile misfit strain. We develop a model to describe the effect of misfit strain on charge ordering in the copper oxygen planes of oxide materials and illustrate some of the low energy stripe nanostructures that can result.

cond-mat.supr-con↗

arXiv2002

Random-field Ising model on complete graphs and trees

We present exact results for the critical behavior of the RFIM on complete graphs and trees, both at equilibrium and away from equilibrium, i.e., models for hysteresis and Barkhausen noise. We show that for stretched exponential and power law distributions of random fields the behavior on complete graphs is non-universal, while the behavior on Cayley trees is universal even in the limit of large co-ordination.

cond-mat.dis-nn↗

arXiv2001

Random manifolds in non-linear resistor networks: Applications to varistors and superconductors

We show that current localization in polycrystalline varistors occurs on paths which are, usually, in the universality class of the directed polymer in a random medium. We also show that in ceramic superconductors, voltage localizes on a surface which maps to an Ising domain wall. The emergence of these manifolds is explained and their structure is illustrated using direct solution of non-linear resistor networks.

cond-mat.stat-mech↗

arXiv2001

Permeability and conductivity of platelet-reinforced membranes and composites

We present large scale simulations of the diffusion constant $D$ of a random composite consisting of aligned platelets with aspect ratio $a/b>>1$ in a matrix (with diffusion constant $D_0$) and find that $D/D_0 = 1/(1+ c_1 x + c_2 x^2)$, where $x= a v_f/b$ and $v_f$ is the platelet volume fraction. We demonstrate that for large aspect ratio platelets the pair term ($x^2$) dominates suggesting large property enhancements for these materials. However a small amount of face-to-face ordering of the platelets markedly degrades the efficiency of platelet reinforcement.

cond-mat.mtrl-sci↗

arXiv2001

Extremal statistics in the energetics of domain walls

We study at T=0 the minimum energy of a domain wall and its gap to the first excited state concentrating on two-dimensional random-bond Ising magnets. The average gap scales as $ΔE_1 \sim L^θf(N_z)$, where $f(y) \sim [\ln y]^{-1/2}$, $θ$ is the energy fluctuation exponent, $L$ length scale, and $N_z$ the number of energy valleys. The logarithmic scaling is due to extremal statistics, which is illustrated by mapping the problem into the Kardar-Parisi-Zhang roughening process. It follows that the susceptibility of domain walls has also a logarithmic dependence on system size.

cond-mat.stat-mech↗

arXiv2001

Minimum spanning trees on random networks

We show that the geometry of minimum spanning trees (MST) on random graphs is universal. Due to this geometric universality, we are able to characterise the energy of MST using a scaling distribution ($P(ε)$) found using uniform disorder. We show that the MST energy for other disorder distributions is simply related to $P(ε)$. We discuss the relationship to invasion percolation (IP), to the directed polymer in a random media (DPRM) and the implications for the broader issue of universality in disordered systems.

cond-mat.stat-mech↗

arXiv2001

Ground state non-universality in the random field Ising model

Two attractive and often used ideas, namely universality and the concept of a zero temperature fixed point, are violated in the infinite-range random-field Ising model. In the ground state we show that the exponents can depend continuously on the disorder and so are non-universal. However, we also show that at finite temperature the thermal order parameter exponent one half is restored so that temperature is a relevant variable. The broader implications of these results are discussed.

cond-mat.dis-nn↗

arXiv2000

Intermittence and roughening of periodic elastic media

We analyze intermittence and roughening of an elastic interface or domain wall pinned in a periodic potential, in the presence of random-bond disorder in (1+1) and (2+1) dimensions. Though the ensemble average behavior is smooth, the typical behavior of a large sample is intermittent, and does not self-average to a smooth behavior. Instead, large fluctuations occur in the mean location of the interface and the onset of interface roughening is via an extensive fluctuation which leads to a jump in the roughness of order $λ$, the period of the potential. Analytical arguments based on extreme statistics are given for the number of the minima of the periodicity visited by the interface and for the roughening cross-over, which is confirmed by extensive exact ground state calculations.

cond-mat.stat-mech↗

arXiv2000

A periodic elastic medium in which periodicity is relevant

We analyze, in both (1+1)- and (2+1)- dimensions, a periodic elastic medium in which the periodicity is such that at long distances the behavior is always in the random-substrate universality class. This contrasts with the models with an additive periodic potential in which, according to the field theoretic analysis of Bouchaud and Georges and more recently of Emig and Nattermann, the random manifold class dominates at long distances in (1+1)- and (2+1)-dimensions. The models we use are random-bond Ising interfaces in hypercubic lattices. The exchange constants are random in a slab of size $L^{d-1} \times λ$ and these coupling constants are periodically repeated along either {10} or {11} (in (1+1)-dimensions) and {100} or {111} (in (2+1)-dimensions). Exact ground-state calculations confirm scaling arguments which predict that the surface roughness $w$ behaves as: $w \sim L^{2/3}, L \ll L_c$ and $w \sim L^{1/2}, L \gg L_c$, with $L_c \sim λ^{3/2}$ in $(1+1)$-dimensions and; $w \sim L^{0.42}, L \ll L_c$ and $w \sim \ln(L), L \gg L_c$, with $L_c \sim λ^{2.38}$ in $(2+1)$-dimensions.

cond-mat.dis-nn↗

arXiv1998

Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices

We show that negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heat can be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.

cond-mat.stat-mech↗

arXiv1998

Quasi-static cracks and minimal energy surfaces

We compare the roughness of minimal energy(ME) surfaces and scalar ``quasi-static'' fracture surfaces(SQF). Two dimensional ME and SQF surfaces have the same roughness scaling, w sim L^zeta (L is system size) with zeta = 2/3. The 3-d ME and SQF results at strong disorder are consistent with the random-bond Ising exponent zeta (d >= 3) approx 0.21(5-d) (d is bulk dimension). However 3-d SQF surfaces are rougher than ME ones due to a larger prefactor. ME surfaces undergo a ``weakly rough'' to ``algebraically rough'' transition in 3-d, suggesting a similar behavior in fracture.

cond-mat↗