Related papers: Monte Carlo comparisons of the self-avoiding walk …
We calculate the correction-to-scaling exponent $\omega_T$ that characterizes the approach to the scaling limit in multicomponent polymer solutions. A direct Monte Carlo determination of $\omega_T$ in a system of interacting self-avoiding…
Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a…
Studying SLE$_{\kappa}$ on $S^2$ provides a new and interesting perspective for the conformality of some 2-dimensional physical models. We prove the existence and some basic properties of the spherical Minkowski content of SLE$_{\kappa}$,…
Numerical studies of fractal curves in the plane often focus on subtle geometrical properties such as their left passage probability. Schramm-Loewner evolution (SLE) is a mathematical framework which makes explicit predictions for such…
We have studied the iso-height lines on the $\mathrm{WO_3}$ surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical…
Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called…
The mating of trees approach to Schramm-Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier-Miller-Sheffield (2014). In this paper we consider the mating of trees…
We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial…
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into…
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site…
A meandric system of size $n$ is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above the real line and the other below the real line. A uniform random meandric system can be…
This paper proves the long-standing open conjecture rooted in chemical physics (Flory (1949)) that the self-avoiding walk (SAW) in the square lattice has root mean square displacement exponent \nu= 3/4. This value is an instance of the…
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of…
Monte Carlo (MC) simulations have been performed to refine the estimation of the correction-to-scaling exponent $\omega$ in the 2D $\varphi^4$ model, which belongs to one of the most fundamental universality classes. If corrections have the…
We develop statistical enumeration methods for self-avoiding walks using a powerful sampling technique called the multicanonical Monte Carlo method. Using these methods, we estimate the numbers of the two dimensional N-step self-avoiding…
We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a consequence we are able to prove that the…
We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the…
We present a Monte Carlo study of the fractal geometry of clusters formed by discrete-time simple random walks (sRW) of $L^2$ steps on a periodic square $L\times L$ lattice. We verify with high precision that the asymptotic behavior of the…
We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a…
We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter p(N) vary with the size N of the box that one is considering, three possibilities arise in…