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Symmetric Random Walks on Regular Tetrahedra, Octahedra and Hexahedra

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Symmetric Random Walks on Regular Tetrahedra, Octahedra and Hexahedra View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by IUPUIScholarWorks Symmetric Random Walks on Regular Tetrahedra, Octahedra and Hexahedra Jyotirmoy Sarkar1 and Saran Ishika Maiti2 1Indiana University{Purdue University Indianapolis, USA 2Visva-Bharati, Santiniketan, India Third Revision: 2017 January 08 Abstract: We study a symmetric random walk on the vertices of three regular polyhedra. Starting from the origin, at each step the random walk moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the distributions, or at least the means and the standard deviations, of the number of steps needed (1) to return to origin, (2) to visit all vertices, and (3) to return to origin after visiting all vertices. We also find the distributions of (i) the number of vertices visited before return to origin, (ii) the last vertex visited, and (iii) the number of vertices visited during return to origin after visiting all vertices. Key Words and Phrases: Absorbing state, Cover time, Dual graph, Eventual transition, One-step transition, Platonic solids, Recursive relation, Return time, Symmetry 2010 Mathematics Subject Classifications: primary 60G50; secondary 60G40 1 Introduction For over two millennia, Platonic solids (also called regular polyhedra) have been regarded as fascinating mathematical objects. A Platonic solid is a three dimensional convex space bounded by congruent regular polygonal faces; and at each of its vertices an equal number of faces meet. Depicted in Fig. 1, are the five Platonic solids|tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron. See Euclid [3], for example, for an elementary explanation as to why there are exactly five Platonic solids. Also see Cromwell [2] for an exposition of their mathematical beauty and symmetry. The Greek philosopher Plato (about 428{347 BCE)|after whom these solids are named|theorized that these five solids correspond to the five classical elements—fire, earth, air, ether and water. In this paper we study symmetric random walks (SRW) on the first three Platonic solids|tetrahedron, octahedron and hexahedron; and we leave the remaining two Platonic solids for later studies. At the outset, we must mention that we are considering regular polyhedra only for aesthetic reasons. However, regularity as such has no bearing on 1Corresponding author email: [email protected] (J. Sarkar) This is the author's manuscript of the article published in final edited form as: Sarkar, J., & Maiti, S. I. (2017). Symmetric Random Walks on Regular Tetrahedra, Octahedra, and Hexahedra. Calcutta Statistical Association Bulletin, 69(1), 110–128. https://doi.org/10.1177/0008068317695974 the results of this paper, because they remain valid even for non-regular solids that are topologically equivalent. We label the vertices of a regular polyhedra according to the following convention: The vertices of a regular tetrahedron are denoted by f0; 1; 2; 3g. It does not matter which vertex gets what label since all pairs of vertices are connected by edges. The vertices of a regular octahedron are denoted by f0; 1; 2; 3; 4; 5g in such a way that three pairs of opposite vertices (0; 5); (1; 4); (2; 3) not directly connected by edges. The vertices of a regular hexahedron are denoted by f0; 1; 2; 3; 4; 5; 6; 7g such that Vertices 1; 2; 3 are adjacent to Vertex 0, Vertices 4; 5; 6 are adjacent to Vertex 7, and four pairs of opposite vertices (0; 7); (1; 6); (2; 5); (3; 4) are not directly connected by edges. See Fig. 2. Let t = 0; 1; 2;::: denote the time epochs when the random walk (RW) moves from one vertex to an adjacent vertex. Let X(t) denote the location of the RW at time t. Without loss of generality, the RW starts at Vertex 0; that is, X(0) = 0.Vertex 0 is called the origin. Thereafter, at each successive epoch, from the current vertex the RW moves to one of the adjacent vertices with equal probability. This is why the RW is called symmetric. Also, as the RW continues on and on, the successive moves are independent. Consequently, fX(t): t ≥ 0g is a discrete-time Markov chain on a discrete state space consisting of all vertices. Because the states are connected and finitely-many, the chain is irreducible. We study the probability distributions of the times until some predefined events hap- pen. Taking a cue from Sarkar (2006), we identified three such events: (1) Return to the origin, (2) Visiting all vertices at least once, and (3) Return to the origin after visiting all vertices. The epochs when these events occur are respectively called the return time TR, ¯ the cover time T , and the additional time to return after visiting all vertices LTR, where ¯ ¯ L = X(T ) 6= 0. Since X(TR) = 0 and X(LTR) = 0, it follows that TR 6= T . Also, we study the probability distributions of NR;L, and LNR defined as follows: (i) NR is the number of non-origin vertices visited until the RW returns to origin; that is, ¯ NR is the number of distinct elements of fX(1);X(2);:::;X(TR − 1)g; (ii) L = X(T ) is the last vertex visited; that is, the location of the RW when all vertices have been visited at least once; and (iii) LNR is the number of vertices visited while the RW returns to origin after visiting all vertices; that is, LNR is the number of distinct elements of ¯ ¯ fX(T );X(T + 1);:::;X(LTR − 1)g. The study of RWs on the vertices of graphs has a rich literature. G¨odeland Jagers [4] studied a RW on a connected graph; and found expressions for the expected recurrence time, first passage time and symmetrized first passage time (also known as commuting time). Letac and Tak´acs[6] studied the limiting probability that an SRW visits a particu- lar vertex of a dodecahedron; and then they related the result to the spectrum of a certain finite-dimensional Banach algebra. In [9] van Slijpe established that, for distance-regular graphs (which include Platonic solids), the mean passage times from one vertex to another and vice versa are equal; and in [10] they extended the result to vertex-transitive graphs. 2 Wildberger [11] described the hypergroup structures (or the character table) associated with any distance transitive graph (which include Platonic solids); and using commuta- tive harmonic analysis, he found the probability of the RW returning to the origin after n steps. Sarkar [8] studied all six questions mentioned above in the context of an asymmet- ric random walk on a polygon. These same questions are answered by Maiti and Sarkar [7] for the SRW on a finite linear path or on a finite cyclic network. Our objective in this paper is two fold: First, to all six questions stated above, we present elementary solutions. Second, we wish to inspire young mathematicians to fill- in some details of derivations and extend this work to study SRW on other interesting graphs. Other researchers might also find our results useful. Let us introduce some terminologies and notation: The distance between two vertices i and j is the number of edges on the shortest path connecting i and j. Let iTj denote the number of steps the RW, starting from Vertex i, takes to reach Vertex j for the first time. That is, iTj = minft > 0 : X(s + t) = j);X(s) = i for some s > 0g with the usual convention of taking 1 as the minimum of the empty set. In fact, the irreducibility of the chain on a finite number of states ensures that iTj so defined is actually finite with probability one. So are all the time variables considered in this paper. For all regular polyhedra, by renumbering the vertices, if necessary, we have TR = 0T0 = 1 + 1T0, and LTR = LT0. Let us denote the mean, the mean square and the variance of iTj, 2 2 respectively by iej = E[iTj], isj = E[iTj ], and ivj = V (iTj) = isj − iej ; where E denotes expectation and V denotes variance. Here are the steps involved in answering each of the six questions about the RW on Platonic solids: (1) We draw an appropriate transition diagram. In this diagram, the nodes represent a triplet consisting of the origin, the vertex where the RW currently is, and the set of vertices the RW has visited already. (2) If the RW can go from one node to another (or itself) in one step, then we draw an arc connecting the nodes; and we specify the transition probability on the arc. (3) Whenever entry into a node signifies that the mission has been accomplished (for whichever purpose the diagram is drawn), we call that node an absorbing state. (4) We eliminate the loops (arcs going from a node to itself) and exchanges (arcs going directly to and fro between two nodes), and simultaneously we revise the conditional transition probabilities of reaching a node that involves one more visited vertex. (5) For the random variable (RV) necessary to answer the question, we obtain the entire probability distribution when possible; otherwise, we report only the mean and the standard deviation (SD). Henceforth, in the context of a particular transition diagram, we reserve the word `vertex' to mean a point where at least three of the edges of a Platonic solid meet, and the word `node' to refer to a triplet of the origin, the current vertex and the set of vertices already visited. Whereas the answers to the six questions for a tetrahedron are rather straight-forward, we still document the detailed solutions in order to develop the notation and to illustrate the techniques.
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