arXiv subjects

Stanislav Smirnov

Publications and source records attributed to Stanislav Smirnov.

At least 19 recordsLinked to original sources

arXiv2025

How long are the arms in DBM?

Diffusion Limited Aggregation and its generalization, Dielectric Breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten's estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in $\mathbb{Z}^2$ and $\mathbb{Z}^3$. The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten's, our argument provides a new proof for Kesten's result both in $\mathbb{Z}^2$ and $\mathbb{Z}^3$.

math.PR↗

arXiv2025

Discrete stress-energy tensor in the loop O(n) model

We study the loop $O(n)$ model on the honeycomb lattice. By means of local non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For $n\in [0,2]$, it gives a new observable satisfying a part of Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy tensor in the continuum, which is known to be a holomorphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of $n=1$ which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stress-energy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge $c=1/2$. Proving the conjecture for other values of $n$ remains a challenge. In particular, this would open a road to establishing the convergence of the interface to the corresponding $\mathrm{SLE}_κ$ in the scaling limit.

math-ph↗

arXiv2023

A generalization of Cardy's and Schramm's formulae

We study critical site percolation on the triangular lattice. We find the difference of the probabilities of having a percolation interface to the right and to the left of two given points in the scaling limit. This generalizes both Cardy's and Schramm's formulae. The generalization involves a new interesting discrete analytic observable and an unexpected conformal mapping.

math.PR↗

arXiv2021

Percolation and $O(1)$ loop model

We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing probabilities. The new approach is more conceptual, less technically demanding, and is amenable to generalizations.

math.PR↗

arXiv2019

Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE

In the second article of this series, we establish the convergence of the loop ensemble of interfaces in the random cluster Ising model to a conformal loop ensemble (CLE) --- thus completely describing the scaling limit of the model in terms of the random geometry of interfaces. The central tool of the present article is the convergence of an exploration tree of the discrete loop ensemble to a branching SLE$(16/3,-2/3)$. Such branching version of the Schramm's SLE not only enjoys the locality property, but also arises logically from the Ising model observables.

math-ph↗

arXiv2019

Conformal invariance of boundary touching loops of FK Ising model

In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces. Some other results in this article are convergence of the exploration process of the loop ensemble (or the branch of the exploration tree) to SLE$(κ,κ-6)$, $κ=16/3$, and convergence of a generalization of this process for $4$ marked points to SLE$[κ,Z]$, $κ=16/3$, where $Z$ refers to a partition function. The latter SLE process is a process that can't be written as a SLE$(κ,ρ_1,ρ_2,\ldots)$ process, which are the most commonly considered generalizations of SLEs.

math-ph↗

arXiv2017

Configurations of FK Ising interfaces and hypergeometric SLE

In this paper, we show that the interfaces in FK Ising model in any domain with 4 marked boundary points and wired--free--wired--free boundary conditions conditioned on a specific internal arc configuration of interfaces converge in the scaling limit to hypergeometric SLE (hSLE). The arc configuration consists of a pair of interfaces and the scaling limit of their joint law can be described by an algorithm to sample the pair from a hSLE curve and a chordal SLE (in a random domain defined by the hSLE).

math-ph↗

arXiv2015

On the critical parameters of the $q\ge4$ random-cluster model on isoradial graphs

The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q\ge 4$, it extends the recent theorem of the two first authors to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if $\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $p_h$ denote the horizontal and vertical edge-weights respectively. We also mention consequences for Potts models.

math.PR↗

arXiv2015

Random curves, scaling limits and Loewner evolutions

In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves, moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.

math-ph↗

arXiv2013

Convergence of Ising interfaces to Schramm's SLE curves

We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter $κ=3$ and $κ=16/3$ respectively.

math-ph↗

arXiv2012

Conformal invariance of lattice models

These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models. They present the theory of discrete holomorphic functions and its applications to planar statistical physics (more precisely to the convergence of fermionic observables). Convergence to SLE is discussed briefly. Many open questions are included.

math.PR↗

arXiv2011

On the scaling limits of planar percolation

We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.

math.PR↗

arXiv2011

The energy density in the planar Ising model

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a fermionic observable and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

math-ph↗

arXiv2011

The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$

We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).

math-ph↗

arXiv2011

Universality in the 2D Ising model and conformal invariance of fermionic observables

It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof of universality and conformal invariance has ever been given, and even physics arguments support (a priori weaker) Möbius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.

math-ph↗

arXiv2011

Discrete complex analysis on isoradial graphs

We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic measures, Green's functions and Poisson kernels to their continuous counterparts. Among other applications, the results can be used to establish universality of the critical Ising and other lattice models.

math.CV↗

arXiv2010

Discrete Complex Analysis and Probability

We discuss possible discretizations of complex analysis and some of their applications to probability and mathematical physics, following our recent work with Dmitry Chelkak, Hugo Duminil-Copin and Clément Hongler.

math-ph↗

arXiv2009

Quasisymmetric distortion spectrum

We give improved bounds for the distortion of the Hausdorff dimension under quasisymmetric maps in terms of the dilatation of their quasiconformal extension. The sharpness of the estimates remains an open question and is shown to be closely related to the fine structure of harmonic measure.

math.CV↗