Random Walks on Barycentric Subdivisions and the Strichartz Hexacarpet M. J. Begue1 D. J. Kelleher2 *G. J. H. Khan3 *D. M. Raisingh4 University of Connecticut Mathematics REU 2011 1Department of Mathematics University of Maryland 2Department of Mathematics University of Connecticut 3Department of Mathematics Boston University 4Department of Mathematics Cornell University YMC at Ohio State University, August 19, 2011 G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 1 / 41 Simplexes and Simplicial Complices A d-dimensional simplex is the convex hull of d + 1 points in general position. A simplicial complex is a collection of simplices with a few conditions about how simplices intersect. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 2 / 41 Simplexes and Simplicial Complices A d-dimensional simplex is the convex hull of d + 1 points in general position. A simplicial complex is a collection of simplices with a few conditions about how simplices intersect. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 2 / 41 Simplicial complexes A simplicial complex is a collection of simplexes with a few added conditions. Condition 1: If a simplicial complex K contains a simplex J, then it contains all of the simplices J 0 ⊂ J Condition 2: If two simplices intersect, they intersect on a subsimplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 3 / 41 Simplicial complexes A simplicial complex is a collection of simplexes with a few added conditions. Condition 1: If a simplicial complex K contains a simplex J, then it contains all of the simplices J 0 ⊂ J Condition 2: If two simplices intersect, they intersect on a subsimplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 3 / 41 Simplicial complexes A simplicial complex is a collection of simplexes with a few added conditions. Condition 1: If a simplicial complex K contains a simplex J, then it contains all of the simplices J 0 ⊂ J Condition 2: If two simplices intersect, they intersect on a subsimplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 3 / 41 Simplicial complexes These are valid simplicial complexes: These are not: G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 4 / 41 Simplicial complexes These are valid simplicial complexes: These are not: G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 4 / 41 What is a barycenter? A barycenter is the center of gravity of a simplex. 1 P In algebraic terms, the barycenter of a simplex J is d+1 i vi where vi are the vertices of J. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 5 / 41 What is a barycenter? A barycenter is the center of gravity of a simplex. 1 P In algebraic terms, the barycenter of a simplex J is d+1 i vi where vi are the vertices of J. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 5 / 41 What is barycentric subdivision? Barycentric subdivision is defined recursively, but its formal definition is a bit tedious. Intuitively, we subdivide all of the faces of a simplex and connect the subdivision of the faces to the barycenter of that simplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 6 / 41 What is barycentric subdivision? Barycentric subdivision is defined recursively, but its formal definition is a bit tedious. Intuitively, we subdivide all of the faces of a simplex and connect the subdivision of the faces to the barycenter of that simplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 6 / 41 What is barycentric subdivision? Barycentric subdivision is defined recursively, but its formal definition is a bit tedious. Intuitively, we subdivide all of the faces of a simplex and connect the subdivision of the faces to the barycenter of that simplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 6 / 41 3-D Barycentric Subdivision This is what the barycentric subdivision of a tetrahedron (3-simplex) looks like G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 7 / 41 Repeated Barycentric Subdivision We can repeatedly barycentrically divide a simplex K. We denote the n-th barycentric subdivision as Kn. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 8 / 41 Adjacent simplices We say that two d-simplexes in Kn are adjacent if they intersect along a d − 1 simplex. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere and Elsewhere)August 19, 2011 9 / 41 Approximating graphs Given Kn, we define a graph by assigning a vertex to each of simplices and an edge if those vertexes are adjacent. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 10 / 41 Approximating Graphs (Cont.) Here is the fourth level approximation: G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 11 / 41 How to Determine Graphs Number of vertices of this graphs grows as ((d + 1)!)n where d is the dimension of the original simplex. So we use an alphabet to create words of length n to specify the cells of Kn. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 12 / 41 How to Determine Graphs Number of vertices of this graphs grows as ((d + 1)!)n where d is the dimension of the original simplex. So we use an alphabet to create words of length n to specify the cells of Kn. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 12 / 41 Words and Alphabets 00 01 11 10 05 15 02 12 04 14 03 13 55 54 24 53 23 25 52 22 50 42 43 33 32 20 51 21 41 44 34 31 40 45 35 30 G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 13 / 41 Words and Alphabets So we have a way of specifying vertices. How do we find the edges? If we think of a good naming scheme, then given a word of length n (ex. 0534134), we can find its neighbors. But more on this later... G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 14 / 41 Words and Alphabets So we have a way of specifying vertices. How do we find the edges? If we think of a good naming scheme, then given a word of length n (ex. 0534134), we can find its neighbors. But more on this later... G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 14 / 41 Contraction and Projection Mappings We can perform contractions. This is the same as tacking on a letter to the front of the word. We also have a projection map from a cell to it's parent cell by deleting the last letter of the word. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 15 / 41 Contraction and Projection Mappings We can perform contractions. This is the same as tacking on a letter to the front of the word. We also have a projection map from a cell to it's parent cell by deleting the last letter of the word. G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 15 / 41 Strichartz Hexacarpet Let the length of the words go to infinity with the same connections for each of the finite length words, we obtain the Strichartz Hexacarpet. But what does this look like? G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 16 / 41 Permutohedron 0.5 0.4 0.3 0.2 0.1 0 0.4 0.2 0 1 −0.2 0.8 0.6 0.4 −0.4 0.2 Cayley graph of Sn with adjacent elements swapped G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 17 / 41 Permutohedron 0.5 0.4 0.3 0.2 0.1 0 0.4 0.2 0 1 −0.2 0.8 0.6 0.4 −0.4 0.2 Cayley graph of Sn with adjacent elements swapped G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 17 / 41 Permutohedron n+1 Take all the permutations the vector (1; 2; : : : ; n + 1) 2 R . Its convex hull of these vectors is the permutohedron G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 18 / 41 Permutohedron n+1 Take all the permutations the vector (1; 2; : : : ; n + 1) 2 R . Its convex hull of these vectors is the permutohedron G. J. H. Khan D. M. RaisinghThe Strichartz (Universities Hexacarpet of Somewhere andAugust Elsewhere) 19, 2011 18 / 41 2nd Level approximation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.4 0.2 0 1 −0.2 0.8 0.6 0.4 −0.4 0.2 0 G.