The Length of an SLE - Monte Carlo Studies
|
|
- Elvin Carter
- 5 years ago
- Views:
Transcription
1 The Length of an SLE - Monte Carlo Studies Department of Mathematics, University of Arizona Supported by NSF grant DMS tgk Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.1/29
2 The Self-Avoiding Walk Take all N step, nearest neighbor walks in the upper half plane, starting at the origin which do not visit any site more than once. Give them the uniform probability measure. Let N. Then let lattice spacing go to zero. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.2/29
3 The Self-Avoiding Walk Take all N step, nearest neighbor walks in the upper half plane, starting at the origin which do not visit any site more than once. Give them the uniform probability measure. Let N. Then let lattice spacing go to zero. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.2/29
4 Natural parameterization of SAW SAW has a natural parameterization. Let W(n) be infinite SAW on unit lattice. Define ω(t) = lim n n ν W(nt), E ω(t) 2 = c t 2ν Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.3/29
5 Natural parameterization of SAW SAW has a natural parameterization. Let W(n) be infinite SAW on unit lattice. Define ω(t) = lim n n ν W(nt), SAW: 100,000 steps, 40 segments of 2,500 steps E ω(t) 2 = c t 2ν Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.3/29
6 Capacity parameterization of SLE SLE is usually parameterized by half-plane capacity. If we divide it into segments of equal change in capacity, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.4/29
7 Capacity parameterization of SLE SLE is usually parameterized by half-plane capacity. If we divide it into segments of equal change in capacity, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.4/29
8 Fractal variation parameterization of SLE If we take the same SLE curves from previous slide and parameterize them by p-variation and divide it into segments of equal variation, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.5/29
9 Fractal variation parameterization of SLE If we take the same SLE curves from previous slide and parameterize them by p-variation and divide it into segments of equal variation, we get Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.5/29
10 Why you want to reparameterize Consider the SAW. It is believed to be SLE(8/3). What does this mean? If we consider the SAW scaling limit curves and SLE curves modulo parameterization, then they have the same distribution. If we use the natural parameterization for SAW and the capacity parameterization for SLE, the parameterized curves do not have the same distribution. If you use capacity to parameterize the SAW, then the SAW curves and the SLE curves should have the same distribution as parameterized curves. Our goal is to reparameterize SLE so it agrees with SAW with its natural parameterization. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.6/29
11 How to reparameterize SLE Use pth variation with p = 1/ν: Let 0 = t n 0 < t n 1 < t n 2 < t n k n = t be sequence of partitions of [0, t]. fvar(γ[0, t]) = lim γ(t n n j ) γ(t n j 1) 1/ν j With ν = 1/2 this is the quadratic variation. For Brownian motion it is non-random and proportional to t. Problem: definition of pth variation depends on parameterization If you compute pth variation for SAW using capacity parameterization and the natural parameterization, you do not get the same result. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.7/29
12 Better definition of fractal variation Let x > 0. Let t i be first time after t i 1 with γ(t i ) γ(t i+1 ) = x Then γ(t j ) γ(t j 1 ) 1/ν = n x 1/ν j where n depends on the curve. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.8/29
13 Better definition of fractal variation Let x > 0. Let t i be first time after t i 1 with γ(t i ) γ(t i+1 ) = x Then γ(t j ) γ(t j 1 ) 1/ν = n x 1/ν j where n depends on the curve. Conjecture: For the scaling limit of a discrete model (LERW, SAW, Ising, percolation), the fractal variation exists and fvar(ω[0, t]) = ct Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.8/29
14 Better definition of fractal variation Let x > 0. Let t i be first time after t i 1 with γ(t i ) γ(t i+1 ) = x Then γ(t j ) γ(t j 1 ) 1/ν = n x 1/ν j where n depends on the curve. Conjecture: For the scaling limit of a discrete model (LERW, SAW, Ising, percolation), the fractal variation exists and fvar(ω[0, t]) = ct Key point: The 1/ν variation of ω[0, t] is not random. (LLN) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.8/29
15 LERW Fractal Variation Compute fractal variation of LERW s of fixed length dt=0.01 dt=0.005 dt=0.002 dt=0.001 LERW fractal variation, 100K steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.9/29
16 LERW Fractal Variation - zoomed Compute fractal variation of LERW s of fixed length dt=0.01 dt=0.005 dt=0.002 dt=0.001 LERW fractal variation, 100K steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.10/29
17 LERW Fractal Variation If dt is too small dt=0.01 dt=0.005 dt=0.002 dt=0.001 dt= dt= dt= LERW fractal variation, 100K steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.11/29
18 SAW Fractal Variation Self-avoiding walks with 1,000,000 steps SAW, fractal variation, 1000K steps dt=0.01 dt=0.005 dt=0.002 dt=0.001 dt= dt= Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.12/29
19 Ising Fractal Variation Interface of Ising model in 1200 by 400 box with Dobrushin boundary conditions Ising interface, fractal variation, 1200 x 400 dt=0.01 dt=0.005 dt=0.002 dt= Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.13/29
20 Percolation Fractal Variation Percolation exploration process in half plane, 4,000,000 steps Percolation, fractal variation, 4000K steps dt=0.01 dt=0.005 dt=0.002 dt=0.001 dt= Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.14/29
21 SLE vs Discrete Models as Parameterized Curves The theorems/conjectures of the form LERW=SLE 2, SAW=SLE 8/3, Ising=SLE 3, percolation=sle 6 are typically statements about paths modulo parametrization or statements where the discrete model is parameterized by capacity. These correspondences should hold if we use the natural parameterization for the discrete model and the fractal variation to parameterize the SLE. To test this we must simulate SLE as well as the discrete models. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.15/29
22 How to simulate SLE Discretize time, but not space: 0 < t 1 < t 2 < < t n = t. Sample BM at these times. Approximate the BM path by a function which agrees at the t i and in between is interpolated in such a way that the Loewner equation has an explicit solution, e.g., c t Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.16/29
23 SLE Simulation subtleties f t = gt 1 is now given by a composition of n explicit conformal maps. Points on SLE are given by z k = f k f 2 f 1 (0) SLE(6) with uniform time discretization - 20,000 steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.17/29
24 SLE Simulation subtleties Better to use a non-uniform time discretization, depending on the BM sample. When step of SLE is too large, sample the BM at a shorter time scale for that time interval using Brownian bridge. SLE(6) with adaptive time discretization - 20,000 steps Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.18/29
25 SLE Simulation subtleties Points on SLE are given by z k = f k f 2 f 1 (0) Time needed to compute n points on SLE is O(n 2 ). There is a trick to make it O(n p ) with p 1.4. z k = f k f lb+1 (f lb f lb+2 f lb+1 ) (f 2b f b+2 f b+1 ) (f b f 2 f 1 )(0) Approximate f i by Laurent series about. Then Laurent series of block function can be computed. Use it when you can. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.19/29
26 SLE vs Discrete For LERW, SAW, percolation in upper half plane, we fix a semicircle of some radius. Compute fractal variation of curve until it hits semicircle. Then look at point that is midway between this point and origin. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.20/29
27 LERW vs SLE(2) in half plane: X Distribution of x-coordinate of midpoint. 1 LERW-chordal half plane SLE(2) SLE(2) using capacity Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.21/29
28 LERW vs SLE(2) in half plane: Y Distribution of y-coordinate of midpoint. 1 LERW - chordal half plane SLE(2) SLE(2) using capacity Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.22/29
29 SAW vs SLE(8/3) in half plane: X Distribution of x-coordinate of midpoint. 1 SAW half plane SLE(8/3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.23/29
30 SAW vs SLE(8/3) in half plane: Y Distribution of x-coordinate of midpoint. 1 SAW SLE(8/3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.24/29
31 Ising vs SLE(3) in half plane: X Distribution of x-coordinate of midpoint Ising SLE(3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.25/29
32 Ising vs SLE(3) in half plane: Y Distribution of y-coordinate of midpoint. 1 Ising SLE(3) Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.26/29
33 Percolation vs SLE(6) in half plane: X Distribution of x-coordinate of midpoint. 1 "perc_0.50_x" "sle_6._0.01_0.50_x" Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.27/29
34 Percolation vs SLE(6) in half plane: Y Distribution of y-coordinate of midpoint. 1 "perc_0.50_y" "sle_6._0.01_0.50_y" Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.28/29
35 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29
36 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Monte Carlo simulations of LERW, SAW, Ising spin interface and percolation exploration support this. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29
37 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Monte Carlo simulations of LERW, SAW, Ising spin interface and percolation exploration support this. For SLE the fractal variation exists. Can you prove this? Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29
38 Conclusions/conjectures/homework For discrete models the fractal variation exists and is proportional to the natural parameterization. Monte Carlo simulations of LERW, SAW, Ising spin interface and percolation exploration support this. For SLE the fractal variation exists. Can you prove this? If we use fractal variation to parameterize SLE, then it agrees as a parameterized curve with the discrete model using its natural time as the parameterization. Stochastic Geometry and Field Theory, KITP, Sept 19, 2006 p.29/29
Numerical simulation of random curves - lecture 1
Numerical simulation of random curves - lecture 1 Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk 2008 Enrage Topical School ON GROWTH
More informationAn Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute. March 14, 2008
An Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute March 14, 2008 Outline Lattice models whose time evolution is not Markovian. Conformal invariance of their scaling
More informationConformal invariance and covariance of the 2d self-avoiding walk
Conformal invariance and covariance of the 2d self-avoiding walk Department of Mathematics, University of Arizona AMS Western Sectional Meeting, April 17, 2010 p.1/24 Outline Conformal invariance/covariance
More informationTesting for SLE using the driving process
Testing for SLE using the driving process Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk Testing for SLE, 13rd Itzykson Conference
More informationAdvanced Topics in Probability
Advanced Topics in Probability Conformal Methods in 2D Statistical Mechanics Pierre Nolin Different lattices discrete models on lattices, in two dimensions: square lattice Z 2 : simplest one triangular
More informationSelf-avoiding walk ensembles that should converge to SLE
Tom Kennedy UC Davis, May 9, 2012 p. 1/4 Self-avoiding walk ensembles that should converge to SLE Tom Kennedy University of Arizona, MSRI Tom Kennedy UC Davis, May 9, 2012 p. 2/4 Outline Variety of ensembles
More informationarxiv:math/ v1 [math.pr] 21 Dec 2001
Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk arxiv:math/0112246v1 [math.pr] 21 Dec 2001 Tom Kennedy Departments of Mathematics and Physics University of Arizona, Tucson, AZ, 85721
More informationGEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE)
GEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE) Triennial Ahlfors-Bers Colloquium Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago 5734 S.
More informationFractal Properties of the Schramm-Loewner Evolution (SLE)
Fractal Properties of the Schramm-Loewner Evolution (SLE) Gregory F. Lawler Department of Mathematics University of Chicago 5734 S. University Ave. Chicago, IL 60637 lawler@math.uchicago.edu December 12,
More information3 Statistical physics models
3 Statistical physics models 3.1 Percolation In Werner s St. Flour lectures he discusses percolation in the first section and then in more detail in section 10. In the article by Kager and Nienhuis percolation
More informationAn Introduction to Percolation
An Introduction to Percolation Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ kozdron/ Colloquium Department of Mathematics & Statistics September 28, 2007 Abstract Percolation was
More informationTowards conformal invariance of 2-dim lattice models
Towards conformal invariance of 2-dim lattice models Stanislav Smirnov Université de Genève September 4, 2006 2-dim lattice models of natural phenomena: Ising, percolation, self-avoiding polymers,... Realistic
More informationPlan 1. Brownian motion 2. Loop-erased random walk 3. SLE 4. Percolation 5. Uniform spanning trees (UST) 6. UST Peano curve 7. Self-avoiding walk 1
Conformally invariant scaling limits: Brownian motion, percolation, and loop-erased random walk Oded Schramm Microsoft Research Weizmann Institute of Science (on leave) Plan 1. Brownian motion 2. Loop-erased
More informationConvergence of loop erased random walks on a planar graph to a chordal SLE(2) curve
Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve Hiroyuki Suzuki Chuo University International Workshop on Conformal Dynamics and Loewner Theory 2014/11/23 1 / 27 Introduction(1)
More informationThe Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution
arxiv:1408.6714v3 [math.pr] 16 Apr 2015 The Smart Kinetic Self-Avoiding Walk and Schramm Loewner Evolution Tom Kennedy Department of Mathematics University of Arizona Tucson, AZ 85721 email: tgk@math.arizona.edu
More informationRandom planar curves Schramm-Loewner Evolution and Conformal Field Theory
Random planar curves Schramm-Loewner Evolution and Conformal Field Theory John Cardy University of Oxford WIMCS Annual Meeting December 2009 Introduction - lattice models in two dimensions and random planar
More information2D Critical Systems, Fractals and SLE
2D Critical Systems, Fractals and SLE Meik Hellmund Leipzig University, Institute of Mathematics Statistical models, clusters, loops Fractal dimensions Stochastic/Schramm Loewner evolution (SLE) Outlook
More information1 Introduction. 2 Simulations involving conformal maps. 2.1 Lowener equation crash course
June 1, 2008. This is very preliminary version of these notes. In particular the references are grossly incomplete. You should check my website to see if a newer version of the notes is available. 1 Introduction
More informationCONFORMAL INVARIANCE AND 2 d STATISTICAL PHYSICS
CONFORMAL INVARIANCE AND 2 d STATISTICAL PHYSICS GREGORY F. LAWLER Abstract. A number of two-dimensional models in statistical physics are conjectured to have scaling limits at criticality that are in
More informationRenormalization group maps for Ising models in lattice gas variables
Renormalization group maps for Ising models in lattice gas variables Department of Mathematics, University of Arizona Supported by NSF grant DMS-0758649 http://www.math.arizona.edu/ e tgk RG in lattice
More informationPlan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM) and percolation (c) Loop-erased random walk (LERW) (d
Percolation, Brownian Motion and SLE Oded Schramm The Weizmann Institute of Science and Microsoft Research Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM)
More informationΩ = e E d {0, 1} θ(p) = P( C = ) So θ(p) is the probability that the origin belongs to an infinite cluster. It is trivial that.
2 Percolation There is a vast literature on percolation. For the reader who wants more than we give here, there is an entire book: Percolation, by Geoffrey Grimmett. A good account of the recent spectacular
More informationSLE and nodal lines. Eugene Bogomolny, Charles Schmit & Rémy Dubertrand. LPTMS, Orsay, France. SLE and nodal lines p.
SLE and nodal lines p. SLE and nodal lines Eugene Bogomolny, Charles Schmit & Rémy Dubertrand LPTMS, Orsay, France SLE and nodal lines p. Outline Motivations Loewner Equation Stochastic Loewner Equation
More informationLocality property and a related continuity problem for SLE and SKLE I
Locality property and a related continuity problem for SLE and SKLE I Masatoshi Fukushima (Osaka) joint work with Zhen Qing Chen (Seattle) October 20, 2015 Osaka University, Σ-hall 1 Locality property
More informationInterfaces between Probability and Geometry
Interfaces between Probability and Geometry (Prospects in Mathematics Durham, 15 December 2007) Wilfrid Kendall w.s.kendall@warwick.ac.uk Department of Statistics, University of Warwick Introduction Brownian
More informationImaginary Geometry and the Gaussian Free Field
Imaginary Geometry and the Gaussian Free Field Jason Miller and Scott Sheffield Massachusetts Institute of Technology May 23, 2013 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian
More informationStochastic Schramm-Loewner Evolution (SLE) from Statistical Conformal Field Theory (CFT): An Introduction for (and by) Amateurs
Denis Bernard Stochastic SLE from Statistical CFT 1 Stochastic Schramm-Loewner Evolution (SLE) from Statistical Conformal Field Theory (CFT): An Introduction for (and by) Amateurs Denis Bernard Chern-Simons
More informationCoarsening process in the 2d voter model
Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 1 / 34 Coarsening process in the 2d voter model Alessandro Tartaglia LPTHE, Université Pierre et Marie Curie alessandro.tartaglia91@gmail.com
More informationExcursion Reflected Brownian Motion and a Loewner Equation
and a Loewner Equation Department of Mathematics University of Chicago Cornell Probability Summer School, 2011 and a Loewner Equation The Chordal Loewner Equation Let γ : [0, ) C be a simple curve with
More informationStochastic Loewner Evolution: another way of thinking about Conformal Field Theory
Stochastic Loewner Evolution: another way of thinking about Conformal Field Theory John Cardy University of Oxford October 2005 Centre for Mathematical Physics, Hamburg Outline recall some facts about
More informationGradient interfaces with and without disorder
Gradient interfaces with and without disorder Codina Cotar University College London September 09, 2014, Toronto Outline 1 Physics motivation Example 1: Elasticity Recap-Gaussian Measure Example 2: Effective
More informationA CURVE WITH NO SIMPLE CROSSINGS BY SEGMENTS
A CURVE WITH NO SIMPLE CROSSINGS BY SEGMENTS CHRISTOPHER J. BISHOP Abstract. WeconstructaclosedJordancurveγ R 2 sothatγ S isuncountable whenever S is a line segment whose endpoints are contained in different
More informationAPPLICATIONS of QUANTUM GROUPS to CONFORMALLY INVARIANT RANDOM GEOMETRY
APPLICATIONS of QUANTUM GROUPS to CONFORMALLY INVARIANT RANDOM GEOMETRY Eveliina Peltola Academic dissertation To be presented for public examination with the permission of the Faculty of Science of the
More informationCFT and SLE and 2D statistical physics. Stanislav Smirnov
CFT and SLE and 2D statistical physics Stanislav Smirnov Recently much of the progress in understanding 2-dimensional critical phenomena resulted from Conformal Field Theory (last 30 years) Schramm-Loewner
More informationNumerical simulation of random curves - lecture 2
Numerical simulation of random curves - lecture 2 Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk 2008 Enrage Topical School ON GROWTH
More informationCONSTRAINED PERCOLATION ON Z 2
CONSTRAINED PERCOLATION ON Z 2 ZHONGYANG LI Abstract. We study a constrained percolation process on Z 2, and prove the almost sure nonexistence of infinite clusters and contours for a large class of probability
More informationRandom curves, scaling limits and Loewner evolutions
Random curves, scaling limits and Loewner evolutions Antti Kemppainen Stanislav Smirnov June 15, 2015 arxiv:1212.6215v3 [math-ph] 12 Jun 2015 Abstract In this paper, we provide a framework of estimates
More informationCurriculum Vitae Thomas Garrett Kennedy
Department of Mathematics University of Arizona Tucson, Arizona 85721, USA Phone: 520-626-0197 e-mail: tgk@math.arizona.edu Home page: www.math.arizona.edu/ tgk Curriculum Vitae Thomas Garrett Kennedy
More informationTHE WORK OF WENDELIN WERNER
THE WORK OF WENDELIN WERNER International Congress of Mathematicians Madrid August 22, 2006 C. M. Newman Courant Institute of Mathematical Sciences New York University It is my pleasure to report on some
More informationarxiv:math-ph/ v2 28 Sep 2006
arxiv:math-ph/0607046v2 28 Sep 2006 Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory Ilya A. Gruzberg The James Franck Institute, The University of Chicago
More informationSLE 6 and CLE 6 from critical percolation
Probability, Geometry and Integrable Systems MSRI Publications Volume 55, 2008 SLE 6 and CLE 6 from critical percolation FEDERICO CAMIA AND CHARLES M. NEWMAN ABSTRACT. We review some of the recent progress
More informationCurriculum Vitae. Thomas Garrett Kennedy
Department of Mathematics University of Arizona Tucson, Arizona 85721, USA Phone: 520-626-0197 e-mail: tgk@math.arizona.edu Home page: www.math.arizona.edu/ tgk Curriculum Vitae Thomas Garrett Kennedy
More informationEE1 and ISE1 Communications I
EE1 and ISE1 Communications I Pier Luigi Dragotti Lecture two Lecture Aims To introduce signals, Classifications of signals, Some particular signals. 1 Signals A signal is a set of information or data.
More informationarxiv: v4 [math.pr] 1 Sep 2017
CLE PERCOLATIONS JASON MILLER, SCOTT SHEFFIELD, AND WENDELIN WERNER arxiv:1602.03884v4 [math.pr] 1 Sep 2017 Abstract. Conformal loop ensembles are random collections of loops in a simply connected domain,
More informationMathematical Research Letters 8, (2001) THE DIMENSION OF THE PLANAR BROWNIAN FRONTIER IS 4/3
Mathematical Research Letters 8, 401 411 (2001) THE DIMENSION OF THE PLANAR BROWNIAN FRONTIER IS 4/3 Gregory F. Lawler 1, Oded Schramm 2, and Wendelin Werner 3 1. Introduction The purpose of this note
More informationExam TFY4230 Statistical Physics kl Wednesday 01. June 2016
TFY423 1. June 216 Side 1 av 5 Exam TFY423 Statistical Physics l 9. - 13. Wednesday 1. June 216 Problem 1. Ising ring (Points: 1+1+1 = 3) A system of Ising spins σ i = ±1 on a ring with periodic boundary
More informationLoewner Evolution. Maps and Shapes in two Dimensions. presented by. Leo P. Kadanoff University of Chicago.
Loewner Evolution Maps and Shapes in two Dimensions presented by Leo P. Kadanoff University of Chicago e-mail: LeoP@UChicago.edu coworkers Ilya Gruzberg, Bernard Nienhuis, Isabelle Claus, Wouter Kager,
More informationarxiv:math/ v1 [math.pr] 27 Mar 2003
1 arxiv:math/0303354v1 [math.pr] 27 Mar 2003 Random planar curves and Schramm-Loewner evolutions Lecture Notes from the 2002 Saint-Flour summer school (final version) Wendelin Werner Université Paris-Sud
More informationCRITICAL BEHAVIOUR OF SELF-AVOIDING WALK IN FIVE OR MORE DIMENSIONS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 2, October 1991 CRITICAL BEHAVIOUR OF SELF-AVOIDING WALK IN FIVE OR MORE DIMENSIONS TAKASHI HARA AND GORDON SLADE ABSTRACT.
More informationCRITICAL PERCOLATION AND CONFORMAL INVARIANCE
CRITICAL PERCOLATION AND CONFORMAL INVARIANCE STANISLAV SMIRNOV Royal Institute of Technology, Department of Mathematics, Stockholm, S10044, Sweden E-mail : stas@math.kth.se Many 2D critical lattice models
More informationPoisson point processes, excursions and stable processes in two-dimensional structures
Stochastic Processes and their Applications 120 (2010) 750 766 www.elsevier.com/locate/spa Poisson point processes, excursions and stable processes in two-dimensional structures Wendelin Werner Université
More informationScaling Theory. Roger Herrigel Advisor: Helmut Katzgraber
Scaling Theory Roger Herrigel Advisor: Helmut Katzgraber 7.4.2007 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff
More informationChapter 4. RWs on Fractals and Networks.
Chapter 4. RWs on Fractals and Networks. 1. RWs on Deterministic Fractals. 2. Linear Excitation on Disordered lattice; Fracton; Spectral dimension 3. RWs on disordered lattice 4. Random Resistor Network
More informationTwo-dimensional self-avoiding walks. Mireille Bousquet-Mélou CNRS, LaBRI, Bordeaux, France
Two-dimensional self-avoiding walks Mireille Bousquet-Mélou CNRS, LaBRI, Bordeaux, France A walk with n = 47 steps Self-avoiding walks (SAWs) Self-avoiding walks (SAWs) A walk with n = 47 steps A self-avoiding
More informationTowards conformal invariance of 2D lattice models
Towards conformal invariance of 2D lattice models Stanislav Smirnov Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising
More informationThe near-critical planar Ising Random Cluster model
The near-critical planar Ising Random Cluster model Gábor Pete http://www.math.bme.hu/ gabor Joint work with and Hugo Duminil-Copin (Université de Genève) Christophe Garban (ENS Lyon, CNRS) arxiv:1111.0144
More informationNumerical Methods with Lévy Processes
Numerical Methods with Lévy Processes 1 Objective: i) Find models of asset returns, etc ii) Get numbers out of them. Why? VaR and risk management Valuing and hedging derivatives Why not? Usual assumption:
More informationSection Arclength and Curvature. (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes.
Section 10.3 Arclength and Curvature (1) Arclength, (2) Parameterizing Curves by Arclength, (3) Curvature, (4) Osculating and Normal Planes. MATH 127 (Section 10.3) Arclength and Curvature The University
More information4: The Pandemic process
4: The Pandemic process David Aldous July 12, 2012 (repeat of previous slide) Background meeting model with rates ν. Model: Pandemic Initially one agent is infected. Whenever an infected agent meets another
More informationA Tour of Spin Glasses and Their Geometry
A Tour of Spin Glasses and Their Geometry P. Le Doussal, D. Bernard, LPTENS A. Alan Middleton, Syracuse University Support from NSF, ANR IPAM: Random Shapes - Workshop I 26 March, 2007 Goals Real experiment
More informationThe dimer model: universality and conformal invariance. Nathanaël Berestycki University of Cambridge. Colloque des sciences mathématiques du Québec
The dimer model: universality and conformal invariance Nathanaël Berestycki University of Cambridge Colloque des sciences mathématiques du Québec The dimer model Definition G = bipartite finite graph,
More informationUniformization and percolation
Uniformization and percolation Itai Benjamini May 2016 Conformal maps A conformal map, between planar domains, is a function that infinitesimally preserves angles. The derivative of a conformal map is
More information16 Singular perturbations
18.354J Nonlinear Dynamics II: Continuum Systems Lecture 1 6 Spring 2015 16 Singular perturbations The singular perturbation is the bogeyman of applied mathematics. The fundamental problem is to ask: when
More informationGradient Percolation and related questions
and related questions Pierre Nolin (École Normale Supérieure & Université Paris-Sud) PhD Thesis supervised by W. Werner July 16th 2007 Introduction is a model of inhomogeneous percolation introduced by
More informationRandom walks, Brownian motion, and percolation
Random walks, Brownian motion, and percolation Martin Barlow 1 Department of Mathematics, University of British Columbia PITP, St Johns College, January 14th, 2015 Two models in probability theory In this
More informationQuantum-Classical Hybrid Monte Carlo Algorithm with Applications to AQC
Quantum-Classical Hybrid Monte Carlo Algorithm with Applications to AQC Itay Hen Information Sciences Institute, USC Workshop on Theory and Practice of AQC and Quantum Simulation Trieste, Italy August
More informationDivergence of the correlation length for critical planar FK percolation with 1 q 4 via parafermionic observables
Divergence of the correlation length for critical planar FK percolation with 1 q 4 via parafermionic observables H. Duminil-Copin September 5, 2018 arxiv:1208.3787v2 [math.pr] 23 Sep 2012 Abstract Parafermionic
More information3.320 Lecture 18 (4/12/05)
3.320 Lecture 18 (4/12/05) Monte Carlo Simulation II and free energies Figure by MIT OCW. General Statistical Mechanics References D. Chandler, Introduction to Modern Statistical Mechanics D.A. McQuarrie,
More informationAdvanced Monte Carlo Methods Problems
Advanced Monte Carlo Methods Problems September-November, 2012 Contents 1 Integration with the Monte Carlo method 2 1.1 Non-uniform random numbers.......................... 2 1.2 Gaussian RNG..................................
More informationDomains and Domain Walls in Quantum Spin Chains
Domains and Domain Walls in Quantum Spin Chains Statistical Interaction and Thermodynamics Ping Lu, Jared Vanasse, Christopher Piecuch Michael Karbach and Gerhard Müller 6/3/04 [qpis /5] Domains and Domain
More informationRandom curves, scaling limits and Loewner evolutions. Kemppainen, Antti
https://helda.helsinki.fi Random curves, scaling limits and Loewner evolutions Kemppainen, Antti 2017-03 Kemppainen, A & Smirnov, S 2017, ' Random curves, scaling limits and Loewner evolutions ', Annals
More informationGlauber Dynamics for Ising Model I AMS Short Course
Glauber Dynamics for Ising Model I AMS Short Course January 2010 Ising model Let G n = (V n, E n ) be a graph with N = V n < vertices. The nearest-neighbor Ising model on G n is the probability distribution
More informationTest 2 Review Math 1111 College Algebra
Test 2 Review Math 1111 College Algebra 1. Begin by graphing the standard quadratic function f(x) = x 2. Then use transformations of this graph to graph the given function. g(x) = x 2 + 2 *a. b. c. d.
More informationAppendix E : Note on regular curves in Euclidean spaces
Appendix E : Note on regular curves in Euclidean spaces In Section III.5 of the course notes we posed the following question: Suppose that U is a connected open subset of R n and x, y U. Is there a continuous
More informationQLE. Jason Miller and Scott Sheffield. August 1, 2013 MIT. Jason Miller and Scott Sheffield (MIT) QLE August 1, / 37
QLE Jason Miller and Scott Sheffield MIT August 1, 2013 Jason Miller and Scott Sheffield (MIT) QLE August 1, 2013 1 / 37 Surfaces, curves, metric balls: how are they related? FPP: first passage percolation.
More informationlim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f),
1 Part I Exercise 1.1. Let C n denote the number of self-avoiding random walks starting at the origin in Z of length n. 1. Show that (Hint: Use C n+m C n C m.) lim n C1/n n = inf n C1/n n := ρ.. Show that
More informationarxiv:math-ph/ v2 6 Jun 2005
On Conformal Field Theory of SLE(κ, ρ) arxiv:math-ph/0504057v 6 Jun 005 Kalle Kytölä kalle.kytola@helsinki.fi Department of Mathematics, P.O. Box 68 FIN-00014 University of Helsinki, Finland. Abstract
More information. Frobenius-Perron Operator ACC Workshop on Uncertainty Analysis & Estimation. Raktim Bhattacharya
.. Frobenius-Perron Operator 2014 ACC Workshop on Uncertainty Analysis & Estimation Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. uq.tamu.edu
More informationNatural parametrization of percolation interface and pivotal points
Natural parametrization of percolation interface and pivotal points Nina Holden Xinyi Li Xin Sun arxiv:1804.07286v2 [math.pr] 3 Apr 2019 Abstract We prove that the interface of critical site percolation
More informationPolymer Solution Thermodynamics:
Polymer Solution Thermodynamics: 3. Dilute Solutions with Volume Interactions Brownian particle Polymer coil Self-Avoiding Walk Models While the Gaussian coil model is useful for describing polymer solutions
More informationConformal field theory on the lattice: from discrete complex analysis to Virasoro algebra
Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra kalle.kytola@aalto.fi Department of Mathematics and Systems Analysis, Aalto University joint work with March 5,
More informationRandom geometric analysis of the 2d Ising model
Random geometric analysis of the 2d Ising model Hans-Otto Georgii Bologna, February 2001 Plan: Foundations 1. Gibbs measures 2. Stochastic order 3. Percolation The Ising model 4. Random clusters and phase
More informationApproximation of Circular Arcs by Parametric Polynomial Curves
Approximation of Circular Arcs by Parametric Polynomial Curves Gašper Jaklič Jernej Kozak Marjeta Krajnc Emil Žagar September 19, 005 Abstract In this paper the approximation of circular arcs by parametric
More informationInterface tension of the 3d 4-state Potts model using the Wang-Landau algorithm
Interface tension of the 3d 4-state Potts model using the Wang-Landau algorithm CP3-Origins and the Danish Institute for Advanced Study DIAS, University of Southern Denmark, Campusvej 55, DK-5230 Odense
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationMulticanonical methods
Multicanonical methods Normal Monte Carlo algorithms sample configurations with the Boltzmann weight p exp( βe). Sometimes this is not desirable. Example: if a system has a first order phase transitions
More informationANALYSIS OF LENNARD-JONES INTERACTIONS IN 2D
ANALYSIS OF LENNARD-JONES INTERACTIONS IN 2D Andrea Braides (Roma Tor Vergata) ongoing work with Maria Stella Gelli (Pisa) Otranto, June 6 8 2012 Variational Problems with Multiple Scales Variational Analysis
More informationPhonon II Thermal Properties
Phonon II Thermal Properties Physics, UCF OUTLINES Phonon heat capacity Planck distribution Normal mode enumeration Density of states in one dimension Density of states in three dimension Debye Model for
More informationMarkov Chains on Countable State Space
Markov Chains on Countable State Space 1 Markov Chains Introduction 1. Consider a discrete time Markov chain {X i, i = 1, 2,...} that takes values on a countable (finite or infinite) set S = {x 1, x 2,...},
More informationComplex Systems Methods 9. Critical Phenomena: The Renormalization Group
Complex Systems Methods 9. Critical Phenomena: The Renormalization Group Eckehard Olbrich e.olbrich@gmx.de http://personal-homepages.mis.mpg.de/olbrich/complex systems.html Potsdam WS 2007/08 Olbrich (Leipzig)
More informationOpen problems from Random walks on graphs and potential theory
Open problems from Random walks on graphs and potential theory edited by John Sylvester University of Warwick, 18-22 May 2015 Abstract The following open problems were posed by attendees (or non attendees
More informationSome Correlation Inequalities for Ising Antiferromagnets
Some Correlation Inequalities for Ising Antiferromagnets David Klein Wei-Shih Yang Department of Mathematics Department of Mathematics California State University University of Colorado Northridge, California
More informationLECTURE 10: Monte Carlo Methods II
1 LECTURE 10: Monte Carlo Methods II In this chapter, we discuss more advanced Monte Carlo techniques, starting with the topics of percolation and random walks, and then continuing to equilibrium statistical
More informationIdentifying supersingular elliptic curves
Identifying supersingular elliptic curves Andrew V. Sutherland Massachusetts Institute of Technology January 6, 2012 http://arxiv.org/abs/1107.1140 Andrew V. Sutherland (MIT) Identifying supersingular
More informationScaling limits of anisotropic random growth models
Scaling limits of anisotropic random growth models Amanda Turner Department of Mathematics and Statistics Lancaster University (Joint work with Fredrik Johansson Viklund and Alan Sola) Overview 1 Generalised
More informationQuasi-Diffusion in a SUSY Hyperbolic Sigma Model
Quasi-Diffusion in a SUSY Hyperbolic Sigma Model Joint work with: M. Disertori and M. Zirnbauer December 15, 2008 Outline of Talk A) Motivation: Study time evolution of quantum particle in a random environment
More informationLiouville quantum gravity as a mating of trees
Liouville quantum gravity as a mating of trees Bertrand Duplantier, Jason Miller and Scott Sheffield arxiv:1409.7055v [math.pr] 9 Feb 016 Abstract There is a simple way to glue together a coupled pair
More informationPoisson random measure: motivation
: motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps
More informationCritical percolation under conservative dynamics
Critical percolation under conservative dynamics Christophe Garban ENS Lyon and CNRS Joint work with and Erik Broman (Uppsala University) Jeffrey E. Steif (Chalmers University, Göteborg) PASI conference,
More informationOn conformally invariant CLE explorations
On conformally invariant CLE explorations Wendelin Werner Hao Wu arxiv:1112.1211v2 [math.pr] 17 Dec 2012 Abstract We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles
More information