Variants of random walks | Discrete geometry | Polygons
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations. In computational physics, a self-avoiding walk is a chain-like path in R2 or R3 with a certain number of nodes, typically a fixed step length and has the property that it doesn't cross itself or another walk. A system of SAWs satisfies the so-called excluded volume condition. In higher dimensions, the SAW is believed to behave much like the ordinary random walk. SAWs and SAPs play a central role in the modeling of the topological and knot-theoretic behavior of thread- and loop-like molecules such as proteins. Indeed, SAWs may have first been introduced by the chemist Paul Flory in order to model the real-life behavior of chain-like entities such as solvents and polymers, whose physical volume prohibits multiple occupation of the same spatial point. SAWs are fractals. For example, in d = 2 the fractal dimension is 4/3, for d = 3 it is close to 5/3 while for d ≥ 4 the fractal dimension is 2. The dimension is called the upper critical dimension above which excluded volume is negligible. A SAW that does not satisfy the excluded volume condition was recently studied to model explicit surface geometry resulting from expansion of a SAW. The properties of SAWs cannot be calculated analytically, so numerical simulations are employed. The is a common method for Markov chain Monte Carlo simulations for the uniform measure on n-step self-avoiding walks. The pivot algorithm works by taking a self-avoiding walk and randomly choosing a point on this walk, and then applying symmetrical transformations (rotations and reflections) on the walk after the nth step to create a new walk. Calculating the number of self-avoiding walks in any given lattice is a common computational problem. There is currently no known formula, although there are rigorous methods of approximation. (Wikipedia).
Self-avoiding walks - Roland Bauerschmidt
Roland Bauerschmidt University of British Columbia; Member, School of Mathematics September 24, 2013 For more videos, visit http://video.ias.edu
From playlist Mathematics
Coding Challenge 162: Self-Avoiding Walk
It's finally time to attempt a Self-Avoiding Walk! In this video, I quickly visualize a simple JavaScript p5.js implementation of a self-avoiding walk. I then tackle the more complex problem of backtracking to find a solution to a space-filling self-avoiding walk. https://thecodingtrain.co
From playlist Recent uploads
Self-avoiding walk in dimension 4 - Roland Bauerschmidt
Self-avoiding walk in dimension 4 - Roland Bauerschmidt Roland Bauerschmidt University of British Columbia; Member, School of Mathematics January 28, 2014 The (weakly) self-avoiding walk is a basic model of paths on the d-dimensional integer lattice that do not intersect (have few interse
From playlist Mathematics
Self-Avoiding Walk and Branched Polymers - John Imbrie
John Imbrie University of Virginia; Member, School of Mathematics March 7, 2011 I will introduce two basic problems in random geometry. A self-avoiding walk is a sequence of steps in a d-dimensional lattice with no self-intersections. If branching is allowed, it is called a branched polyme
From playlist Mathematics
Hugo Duminil-Copin - 4/4 The Self-Avoiding Walk Model
The course will focus on rigorous results for the self-avoiding walk model on lattices, with a special emphasis on low-dimensional ones. The model is defined by choosing uniformly at random among random walk paths starting from the origin and without self-intersections. Despite its simple
From playlist Hugo Duminil-Copin - The Self-Avoiding Walk Model
How to Be Charming When Talking About Yourself
It’s sometimes assumed that talking too much about ourselves is rude; and asking questions of others is polite and charming. But the distinction is not quite so simple. There are far better and worse ways of speaking about ourselves. We end up charming when we dare to reveal our vulnerabil
From playlist SELF
Hugo Duminil-Copin - 1/4 The Self-Avoiding Walk Model
The course will focus on rigorous results for the self-avoiding walk model on lattices, with a special emphasis on low-dimensional ones. The model is defined by choosing uniformly at random among random walk paths starting from the origin and without self-intersections. Despite its simple
From playlist Hugo Duminil-Copin - The Self-Avoiding Walk Model
Two-dimensional Self Avoiding Walks - Mireille Bousquet-Melou (2014)
Slides for this talk: https://docs.google.com/viewer?url=http://www.msri.org/workshops/616/schedules/14374/documents/1552/assets/17052 Abstract: Exactly solvable classes of self-avoiding walks Mireille Bousquet-Mélou Université Bordeaux 1 A walk on a lattice is self-avoiding if it never
From playlist Mathematics
What is a Walk? | Graph Theory
What is a walk in the context of graph theory? That is the subject of today's math lesson! A walk in a graph G can be thought of as a way of moving through G, where you start at any vertex in the graph, and then move to other vertices through the edges in the graph. In a walk, you are allo
From playlist Graph Theory
Mireille Bousquet Melou - Exactly solvable models of self-avoiding walks
slides for this talk: https://docs.google.com/viewer?url=http://www.msri.org/workshops/616/schedules/14374/documents/1552/assets/17052 Connections For Women: Discrete Lattice Models In Mathematics, Physics, And Computing January 12, 2012 - January 13, 2012 January 13, 2012 (04:20 PM PST
From playlist Mathematics
Conformally invariant measures on paths and loops – Gregory Lawler – ICM2018
Plenary Lecture 5 Conformally invariant measures on paths and loops Gregory Lawler Abstract: There has been incredible progress in the last twenty years in the study of fractal paths and fields that arise in planar statistical physics. I will give an introduction to the area and discuss
From playlist Plenary Lectures
15-year-old did mathematical research?! What was my thinking process back then?
Mathematical research can be accessible. This video series somewhat faithfully records the thinking process of a 15-year-old in his mathematical research. Although it turned out that someone had made a similar discovery, this independent and nontrivial discovery still stands to prove that
From playlist Miscellaneous
Gady Kozma: Noncommutative lace expansion
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Probability and Statistics
Dmitry Ioffe: Low temperature interfaces and level lines in the critical prewetting regime
Abstract: Complete wetting in the context of the low temperature two-dimensional Ising model is characterized by creation of a mesoscopic size layer of the "-" phase above an active substrate. Adding a small positive magnetic field h makes "-"-phase unstable, and the layer becomes only mic
From playlist Probability and Statistics
Séminaires du Magistère de Mathématiques 2016 - Vincent Beffara
Quelques questions autours des marches aléatoires 17 mars 2016
From playlist Séminaires du Magistère de Mathématiques
A particularly awful aspect of self-hatred is that we may not even be aware that we are suffering from it. Here is a list of questions to help us diagnose an absence of self-compassion. Sign up to our mailing list to receive 10% off your first order with us: https://r1.dotdigital-pages.com
From playlist SELF