Critical percolation: the expected number of clusters in a rectangle

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.     

Connection probabilities and RSW‐type bounds for the two‐dimensional FK Ising model

We prove Russo‐Seymour‐Welsh‐type uniform bounds on crossing probabilities for the FK Ising (FK percolation with cluster weight q = 2) model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model [24], which allows us to get precise estimates on boundary connection probabilities. We stay in a discrete setting; in particular, we do not make use of any continuum limit, and our result can be used to derive directly several noteworthy properties—including some new ones—among which are the fact that there is no infinite cluster at criticality, tightness properties for the interfaces, and the existence of several critical exponents, in particular the half‐plane, one‐arm exponent. Such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the Ising spin field [6] and deriving polynomial bounds for the mixing time of the Glauber dynamics at criticality [17]. © 2011 Wiley Periodicals, Inc.     

Numerical Scaling Invariance Applied to the van der Pol Model

In this study we use the van der Pol model to explain a novel numerical application of scaling invariance. The model in point is not invariant to a scaling group of transformations, but by introducing an embedding parameter we are able to recover it from an extended model which is invariant to an extended scaling group. As well known, within a similarity analysis we can define a family of solutions from a computed one, so that the solution of a target problem can be obtained by rescaling the solution of a reference problem. The main idea is to use scaling invariance and numerical analysis to find a reference problem easier to solve, from a numerical viewpoint, than the target problem. This allows us to save human efforts and computational resources every time we have to solve a challenging problem. We test our approach using three stiff solvers available within the most recent releases of MATLAB. Independently from the solver used, by employing the described scaling invariance we are able to significantly reduce the computational cost of the numerical solution of the van der Pol model.      

Conformally Invariant Operators,Differential Forms,Cohomology and a Generalisation of Q-Curvature

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator, gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalizes the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generalize in a natural way Branson's Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to the tractor bundles for the Cartan normal conformal connection. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order n in even dimension n. In the case of unweighted (or true) forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.     

An integral invariant from the view point of locally conformally Kähler geometry

In this article we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980s. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifold.     

Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation 1 $\frac{{\partial f}}{{\partial t}} = z\frac{{\zeta (t) + z}}{{\zeta (t) - z}}\frac{{\partial f}}{{\partial z}}$ , with boundary valuesf(z,0)=z, in the rangez ∈U= {w ∈ ? : ?w? < 1},t≤0. We choose ζ(t):=B(?2t), where B(t) is Brownian motion on ? $ \mathbb{U} The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the L?wner differential equation

(1)

, with boundary valuesf(z,0)=z, in the rangez ∈U= {w ∈ ℂ : •w• < 1},t≤0. We choose ζ(t):=B(−2t), where B(t) is Brownian motion on ∂ starting at a random-uniform point in ∂ . Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂ has the same law as that of the pathf(ζ(t),t) (wheref(z,t) is extended continuously to ∂ ) ×(−∞,0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters. Research supported by the Sam and Ayala Zacks Professorial Chair.     

Conformal covariance of the Abelian sandpile height one field

We study the scaling limit for the height one field of the two-dimensional Abelian sandpile model. The scaling limit for the covariance having height one at two macroscopically distant sites, more generally the centred height one joint moment of a finite number of macroscopically distant sites, is identified and shown to be conformally covariant. The result is based on a representation of the height one joint intensities that is close to a block-determinantal structure.     

On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

Exponential Decay for the Near-Critical Scaling Limit of the Planar Ising Model

We consider the Ising model at its critical temperature with external magnetic field ha^15/8 on the square lattice with lattice spacing a . We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0 . As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1 , the mass (inverse correlation length) is of order h^8/15 as h ↓ 0 ; for the Euclidean field, it equals exactly Ch^8/15 for some C . Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

On Algorithms Invariant to Nonlinear Scaling with Inexact Searches

Among the researches of unconstrained optimization invarianey to nnolinear scaling is an interesting subject. But the discussions which have been so far made were all under the assumption of exact line searches. Hence there are some essential deficiency in theory and practice. In this paper, using more generalized concept of invariance, the invariant algorithms not depending on the accuracy of line searches are established for the model presented by Boland et al. in [2].     

Conservation laws for conformally invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.     

The nested simple conformal loop ensembles in the Riemann sphere

Simple conformal loop ensembles (CLE) are random collections of simple non-intersecting loops that are of particular interest in the study of conformally invariant systems. Among other things related to these CLEs, we prove the invariance in distribution of their nested full-plane versions under the inversion \(z \mapsto 1/z\).     

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D₂ ? D₁ to any quadratic differential on D₁ with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (?q)² for a singular harmonic function q on D₁.As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].     

Diffusion of Loops

We study a lattice model that is closely related to the Ising model and can be regarded as describing diffusion of loops in two dimensions. The time development is given by a transfer matrix for a random surface model on a three-dimensional lattice. The transfer matrix is indexed by loops and is invariant under a group of motions in the loop space. The eigenvalues of the transfer matrix are calculated in terms of the partition function and the correlation functions of the Ising model.     

The conformal Killing equation on forms—prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on (k−?)-forms for various integers ?. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.     

Conformally invariant systems of differential operators

Kostant's theory of conformally invariant differential operators on certain homogeneous spaces is generalized to cover conformally invariant systems of endomorphism-valued differential operators. In particular, the connection discovered by Kostant between conformally invariant operators and highest weight vectors in generalized Verma modules is extended.     

Invariant description of ?? N?1 sigma models

We propose an invariant formulation of completely integrable ?? ^N?1 Euclidean sigma models in two dimensions defined on the Riemann sphere S². We explicitly take the scaling invariance into account by expressing all the equations in terms of projection operators, discussing properties of the operators projecting onto one-dimensional subspaces in detail. We consider surfaces connected with the ?? ^N?1 models and determine invariant recurrence relations, linking the successive projection operators, and also immersion functions of the surfaces.     

Yang-mills equations in 4-dimensional conformally connected manifolds

In this paper on the simplest examples of compact 4-dimensional conformally connected manifolds (real quadrics in a 5-dimensional projective space) we show that the only invariant, which is quadratic with respect to the curvature Φ of the connectivity, is the Yang-Mills functional ε |tr (*Φ Λ Φ)|. We do not know whether the 4-form |tr (*Φ Λ Φ)| is invariant in any 4-dimensional conformally connected manifolds.     

Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

A new scaling invariant regularity criterion for the 3D MHD equations in terms of horizontal gradient of horizontal components

We prove a new scaling invariant regularity criterion for the 3D MHD equations via horizontal gradient of horizontal components of weak solutions. This result improves a recent work by Ni et al. (2012), in the sense that the assumption on the horizontal gradient of the vertical components is removed. As a byproduct, a scaling invariant regularity criterion involving vertical components of vorticity and current density is also obtained.