Infinite Contour Research Articles

Ising percolation in the hyperbolic plane

We study infinite “+” or “−” clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph G with finite vertex degree. If the critical percolation probability pcsite for the independent identically distributed (IID). Bernoulli site percolation on G is less than 12, we find an explicit region for the coupling constant of the Ising model such that there are infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters, while the random cluster representation of the Ising model has no infinite 1-clusters. If pcsite>12, we obtain a lower bound for the critical probability in the random cluster representation of the Ising model in terms of pcsite. We also obtain an explicit region for the coupling constant when the XOR Ising model (the product of two IID Ising models) does not have a unique infinite contour a.s. and an explicit region for the coupling constant when the XOR Ising model has infinitely many infinite “+”-clusters and infinitely many infinite “−”-clusters.

Constrained percolation in two dimensions

We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.

Stretching, twisting and supercoiling in short, single DNA molecules

We had combined the Neukirch–Marko model that describes the extension, torque and supercoiling in single, stretched and twisted DNA of infinite contour length, with a form of the free energy suggested by Sinha and Samuels to describe short DNA, with contour length only a few times the persistence length.We find that the free energy of the stretched but untwisted DNA, is significantly modified from its infinitely length value and this in turn modifies significantly the torque and supercoiling. We show that this is consistent with short DNA being more flexible than infinitely long DNA. We hope our results will stimulate experimental investigation of torque and supercoiling in short DNA.

Finite-size corrections for confined polymers in the extended de Gennes regime.

Theoretical results for the extension of a polymer confined to a channel are usually derived in the limit of infinite contour length. But experimental studies and simulations of DNA molecules confined to nanochannels are not necessarily in this asymptotic limit. We calculate the statistics of the span and the end-to-end distance of a semiflexible polymer of finite length in the extended de Gennes regime, exploiting the fact that the problem can be mapped to a one-dimensional weakly self-avoiding random walk. The results thus obtained compare favorably with pruned-enriched Rosenbluth method (PERM) simulations of a three-dimensional discrete wormlike chain model of DNA confined in a nanochannel. We discuss the implications for experimental studies of linear λ-DNA confined to nanochannels at the high ionic strengths used in many experiments.

Orthogonal polynomials in the normal matrix model with a cubic potential

We consider the normal matrix model with a cubic potential. The model is ill-defined, and in order to regularize it, Elbau and Felder introduced a model with a cut-off and corresponding system of orthogonal polynomials with respect to a varying exponential weight on the cut-off region on the complex plane. In the present paper we show how to define orthogonal polynomials on a specially chosen system of infinite contours on the complex plane, without any cut-off, which satisfy the same recurrence algebraic identity that is asymptotically valid for the orthogonal polynomials of Elbau and Felder.The main goal of this paper is to develop the Riemann–Hilbert (RH) approach to the orthogonal polynomials under consideration and to obtain their asymptotic behavior on the complex plane as the degree n of the polynomial goes to infinity. As the first step in the RH approach, we introduce an auxiliary vector equilibrium problem for a pair of measures (μ1,μ2) on the complex plane. We then formulate a 3×3 matrix valued RH problem for the orthogonal polynomials in hand, and we apply the nonlinear steepest descent method of Deift–Zhou to the asymptotic analysis of the RH problem. The central steps in our study are a sequence of transformations of the RH problem, based on the equilibrium vector measure (μ1,μ2), and the construction of a global parametrix.The main result of this paper is a derivation of the large n asymptotics of the orthogonal polynomials on the whole complex plane. We prove that the distribution of zeros of the orthogonal polynomials converges to the measure μ1, the first component of the equilibrium measure. We also obtain analytical results for the measure μ1 relating it to the distribution of eigenvalues in the normal matrix model which is uniform in a domain bounded by a simple closed curve.

Plateau’s problem for infinite contours

Plateau’s problem for infinite contours

Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernándezet al.(1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

Random dynamics and thermodynamic limits for polygonal Markov fields in the plane

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

Cyclicity of Infinite Contour around Certain Reversible Quadratic Center

The paper deals with quadratic vector fields on the Poincairé sphere with center of the reversible kind.x=y(1−2x),y=−x+ay2+bx2. In this case there is a 2-cycle with hyperbolic vertices at infinity. We show that the cyclicity of this 2-cycle is 2 or 3. The last possibility takes place along some curve in the (a, b) plane.

Hyperasymptotics for integrals with finite endpoints

Berry & Howls (1991) (hereinafter called BH) refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp {– kf ( z )} along doubly infinite contours in the complex plane passing over saddlepoints of f ( z ). Here we derive analogous results for integrals with integrands of a similar form, but whose local expansions in powers of 1/ k are made about the finite endpoints of semi-infinite contours of integration. We treat endpoints where f ( z ) behaves locally linearly or quadratically. Generically, local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f ( z ). We derive ‘resurgence relations’ which express the original integral exactly as a truncated endpoint expansion plus a remainder, involving the global saddle structure of f ( z ) via integrals through certain ‘adjacent’ saddles. The saddles adjacent to the endpoint are determined by a topological rule. If the least term of the endpoint expansion is the N 0 ( k ) th, summing to here calculates the endpoint integral up to an error of approximately exp ( – N 0 ( k )). We develop a scheme, involving iteration of the new resurgence relations with a similar one derived in BH, which can reduce this error down to exp( – 2.386 N 0 ( k )). This ‘hyperasymptotic’ formalism parallels that of BH and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane, provided that it does not coincide with other saddles. We illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc( x ) and a constructed ‘incomplete’ Airy function.

Does contour dynamics go singular?

It has recently been claimed that the ‘‘contour-dynamics’’ model of two-dimensional inviscid vortex dyanmics spontaneously develops tangent-slope discontinuities, implying infinite contour curvature, within a finite time. The claim was made on the basis of numerical experimentation on three examples of vortex-patch merging, for one of which details were presented. This note presents contrary numerical evidence for the same example, strongly suggesting that no such singularity develops in the time claimed. The results are generated using overwhelmingly high spatial and temporal resolution. Comparisons with coarser-resolution calculations verify that the effects of resolution have been reduced to an insignificant level at the locations in question. It appears likely that contour smoothness persists for all finite time.

Simple Analytic Proof of the Prime Number Theorem

(1980). Simple Analytic Proof of the Prime Number Theorem. The American Mathematical Monthly: Vol. 87, No. 9, pp. 693-696.