Off-critical lattice models and massive SLEs
We suggest how versions of Schramm's SLE can be used to describe the scaling limit of some off-critical 2D lattice models. Many open questions remain.
math-ph↗
arXiv subjects
Stanislav Smirnov
Publications and source records attributed to Stanislav Smirnov.
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 in principle should provide a new approach to establishing conformal invariance of percolation.
math.PR↗
Critical percolation in the plane
We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy's formula. Then we prove existence, uniqueness, and conformal invariance of the continuum scaling limit.
math.PR↗
Dimension of quasicircles
We introduce canonical antisymmetric quasiconformal maps, which minimize the quasiconformality constant among maps sending the unit circle to a given quasicircle. As an application we prove Astala's conjecture that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$.
math.CV↗
Non-uniform hyperbolicity in complex dynamics
We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and non-atomic conformal measure with exponent equal to the Hausdorff dimension of the Julia set. There is an absolutely continuous invariant measure with respect to this conformal measure. The Minkowski dimension of the Julia set is strictly less than 2. Either the Julia set is the whole sphere, or the dynamics is unstable. For such polynomials and Blaschke products the Julia set is conformally removable. The connected components of the boundary of invariant Fatou components are locally connected. Finally, we derive a conformal analogue of Jakobson-Benedicks-Carleson theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points in the Mandelbrot set with respect to the harmonic measure. Some of the results extend to the multimodal maps of an interval.
math.DS↗
Towards conformal invariance of 2D lattice models
Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to Schramm-Loewner Evolution.
math-ph↗
Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.
math-ph↗
Critical exponents for two-dimensional percolation
We show how to combine Kesten's scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov's proof of Cardy's formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.
math.PR↗