RANDOM WALKS AND PERCOLATION: AN ANALYSIS OF CURRENT RESEARCH ON MODELING NATURAL PROCESSES

RANDOM WALKS AND PERCOLATION: AN ANALYSIS OF CURRENT RESEARCH ON MODELING NATURAL PROCESSES AARON ZWIEBACH Abstract. In this aer we will analyze research that has been recently done in the field of discrete mathematics, secifically relating to modeling natural rocesses. We examine, summarize, and exlain the roofs related to the following natural rocesses: Percolation and Random Walks. We resent the main results of two aers and rovide the roofs to demonstrate how combinatorics, exected value, and generating functions, among other tools, can be used to model such natural rocesses using discrete mathematics.. Introduction Both aers resented in this revew were written by: Hochstadt, Padilla, and Zwiebach, students of Mathematics at the Massachusetts Institute of Technology. The research aers are of interest because they tackle the issues of modeling natural rocesses in nature via mathematics. The two main methods of analysis are combinatorial roofs and emirical analyses. The two random rocesses discussed in the aers are: Random Walks and Percolation. We rovide attachments containing the original aer with the comlete set of results, but for the uroses of this survey aer, we focus on the main results of each aer- those that deal with finding roof based models of the rocesses. More secifically the structure of the aer will be as such: () Introduction Descrition of Random Walks in Z Descrition of Percolation. () Proofs Main result from Analysis of Random Walks in Z Main Result from Percolation (3) Discussion/Conclusion.. Random Walks. In this articular verison of Random Walks, we examine a "walker" taking random stes in Z,thetwodimensionallattice. Our walker takes a ste at each discrete time interval t and must go in in one of the four cardinal directions (left, right, u, or down.) More recisely, if our walker finds themselves at location (x, y) at time t, thenattimet +, they will be at either (x, y ± ) or (x ±,y). Sincethisisarandomrocess we stiulate that the walker goes in any of the four cardinal directions with robability. Since the walker cannot stay in the same sot, they take a 4 ste at each time interval with robability.

AARON ZWIEBACH In the aer that we are reviewing, we exlore the dynamic between our walker and a series of boundary oints. We define a boundary oint as a oint, (x 0,y 0 ),thatoncereachedendstherandomwalk.imagineourwalker finds themsemselves initialized at the center of a square boundary and begins the random walk, with what robability would they reach the boundary at any given time ste, t? What is the walker s exected time until "absorbtion"? What is the likelihood that our walker exits at one articular oints as oosed to the other? How could we use comuter simulations to answer the revious question? How do the revious results vary when we vary the shae of the boundary? These are all natural questions that arise when examining the question of the walker in a bounded area. In Analysis of Random Walks on Z, Hochstadt, Padilla, and Zwiebach examine all of these questions. Given our desire to examine methods of modeling natural rocesses using discrete math, we will resent the roof for one of the main results of the aer. Secifically we will look at the behavior of a walker intialized in a 5x5 square boundary, and exlore the question of the walker s robability of absortion at any given time ste and their exected time until absortion. We cite the following theorems: Theorem. Given a 5x5 array and an initial location (0, 0), therobability that our walker will be absorbed by the boundary at any time ste n is given by: P [Absorbtion] = k Where k is given by a function F (n) that mas the set of even numbers and the set of odd numbers to the set of natural numbers: k = F (n) = ( n if n is even n if n is odd Theorem. The exected time until absorbtion for a walker starting at the center of a 5x5 lattice is 9 time stes. The reader may now be asking themselves: why did we choose to resent aroofonsuchaarticularshaeasa5x5square boundary? As we shall see in the roof, the authors not only rovide us with the two theorems listed above, they also rovide us with a general yet owerful framework for how to answer this question for any shae or dimension of boundary. For that reason we may look at their method for this articular instance and extraolate this method to different instances with different dimensions... Percolation. The second natural rocess that we will examine in this summary aer is Percolation. In nature ercolation is the rocess by which aliquidmovesandisfilteredthroughaorousmaterial. Intheiraer, Hochstadt, Padilla, and Zwiebach, bring the examle of rain falling onto a atch of soil. As the rainwater lands on the surface, it sees through the naturally formed aths in the soil until it is stoed. This rocess is called ercolation and it is the way that the roots of lants are watered!

RANDOM WALKS AND PERCOLATION 3 So, how can we model ercolation using discrete mathematics? We do this by using grahs, secifically trees. We examine the aths accessible from a root vertex. We say that a node is accessible if there is a ath comrised of "oen edges" leading to it. At this oint we should define our terms more recisely. We imagine that we have a root vertex, v, of a tree. We also imagine that this tree has degree n, which means that there are n edges eminating from it leading to n nodes. Futhermore, each one of those n nodes has n edges emanating from it leading to n distinct nodes (in a tree there are no cycles.) If we assume this tree is infinite then this rocess goes on forever. For the uroses of ercolation we assume that we have this infinite tree, and we start at an arbitrarily chosen root vertex. We "oen" each adjacent edge with robability and "close" it with robability. Ifwecloseanedge, the we sto the rocess along that ath and consider all of its descendents unreachable. If we oen an edge then we reeat the rocess for all of its descendents. The rocess can either sto when we close all candidate edges or it can go on ad infinitum. We call a tree in which we oen edges with robability, aercolationgrah with edge robability. Wemeasurethe "cluster" sizeofourgrahby counting all the reachable nodes via oen edges. Figure gives an examle of this rocess: Figure. In this grah, blue edges are oen and red edges are closed. The vertex v is a member of a cluster of size In the aer we exlore the behaviors on a secific kind of tree, trees of valence d. Valence trees are those in which every edge is adjacent to exactly d other edges. Figure gives an examle of a tree of valence 3:

4 AARON ZWIEBACH Figure. The tree of of valence 3 We resent the main result from: Percolation. Namely, weshowhowto derive a closed form function for the size of a cluster formed in a ercolation tree given a valence d and an oen edge robability. Theorem 3:In the grah of valence d with oen edge robability, the exected size of a cluster containing an arbitrary vertex v is: + >< if [0, ) d+ d E[ cluster size ] = if [ d >:, ] 0 otherwise.3. Summary of Introduction. In this section we have resented two roblems that resemble natural rocesses. We have shown how they are relatable to discrete mathematics and by extension claimed that as such we can use combinatorics and robability to generate interesting results. In the next section we resent two roofs from recent aers that we feel reresent well this method of analysis and serve well as an examle for those looking to model their own natural rocesses through discrete mathematics.. Walker in a 5x5 Boundary In this section we cite one of the main results from Analysis of Random Walks in Z by Hochstadt, Padilla, and Zwiebach. The result shown is quoted exactly as it aears in their aer. We resent the result here as it aears in their aer because of the clarity with which they resent their concet would be difficult to relicate. In this roof they derive through induction the robability that the walker is absorbed at any time ste t in a 5x5 boundary. Additionally, they rove that the exected time until absortion is 9 times stes. As we stated in the introduction, the roofs are of articular interest because they are based on robabilistic and combinatorial arguments, rather than on simulations and analysis of the data. Proof as resented in Section 4 of the referred aer: We have a random walker moving in a two-dimensional lane. At any given time ste, the walker can move in any of the four cardinal directions (u,

RANDOM WALKS AND PERCOLATION 5 down, left, and right) each with robability 4. In order to calculate an exlicit formula for the exected time until the walker is absorbed, we first need to derive a formula for the robability that the walker is absorbed by the boundary at any time t. We lot for a few time stes t the robability that our walker will occuy a given location in the lattice. For our uroses we will consider a 5x5 array (Table.) We initialize our walker in the center of the lane, (0, 0) and consider the square boundary with corners: (, ),...(, ),...(, ),...(, ). Table. Position of the walker at t =0 X At first the walker is ositioned at the center of the array and the first ste it takes can either be u, down, left or right, each with robability 4 (Shown in Table.) Table. Possible ositions of walker at t = 4 4 4 4 Given the initial location of the walker and the size of the array, it is imossible for the boundary to absorb the walker at t =.Attimet =,though, the walker can could be absorbed at four distinct oints on the boundary, arriving at each with robability (Seen in Table 3.) 6 Table 3. Possible ositions of walker at t = 6 6 4 6 6 At this oint we encounter our first non-zero robability of the walker being absorbed by the boundary. We calculate the robability by summing the individual robabilities of our walker occuying some oint along the

6 AARON ZWIEBACH boundary. In this case the robability that the walker is absorbed at time t =is + + + =. 6 6 6 6 4 Table 4. Possible ositions of the walker at t =3 3 3 3 3 3 3 3 3 We exressly lot out the ossible ositions that the walker can occuy at t =3in Table 4. We invite the reader to create their own lots for the next few time stes t =4, 5, 6, 7..., calculatingateachointtherobabilityof absortion. This will hel the reader see the attern that forms, and understand the basis for Theorem. Theorem 3. Given a 5x5 array and an initial location (0, 0), therobability that our walker will be absorbed by the boundary at any time ste n is given by: P [Absorbtion] = k Where k is given by a function F (n) that mas the set of even numbers and the set of odd numbers to the set of natural numbers: ( n if n is even k = F (n) = n if n is odd Proof. This will be a roof by induction. Inductive Hyothesis: Forn the board can have two configurations, one for even number stes and one for odd number stes. The configuration for an even time ste is given by Table 5. Table 5. Possible ositions of walker at even time ste t = n Even 3 k k k 3 k k 3 k k k 3 k The configuration for an odd time ste is given by Table 6. Base Cases: We have two base cases. The first case is the configuration of the board at time t =0(the configuration is given by Table.) The robability that the walker is absorbed by the boundary is 0, while the the robability that the walker remains on the lattice is. The second base case is the configuration of the board at time t =. The configuration of the board at t =is given by Table. The robability

RANDOM WALKS AND PERCOLATION 7 Table 6. Possible ositions of walker at odd time ste t = n Odd 4 k 4 k 4 k k 4 k k k 4 k k 4 k 4 k 4 k that the walker is absorbed by the boundary is 0 and the robability that the walker remains on the lattice is. Inductive Ste: We begin by considering an odd time ste t = n +, we note that k = F (n +) = F (n). By our inductive hyothesis board configuration at time n is given by Table 5. The general configuration of the board at time t = n +is given by Table 7.: Table 7. Possible ositions of walker at odd time ste t = n + A B H I C L J G K D F E The robability that the article occuies osition A, is given by P (A). We note that: P (A) =P (H) =P (G) =P (F )=P (E) =P (D) =P (C) =P (B) = 4 k = 4 k Additionally: P (L) =P (I) =P (J) =P (k) = 4 ( k + k + k )= 3 k = k We note that the individual robabilities of each square corresond to those in our inductive hyothesis. By summing all the ositions on the boundary, (A, B, C, D, E, F, G, H), wecancalculatetherobabilityofbeingabsorbed by the border at t = n +: P [Absorbtion] = ( 4 k )= k We now consider odd time ste t = n and even time ste t = n. Wenote that F (n ) = k and F (n) =k +. By our inductive hyothesis the board configuration at t = n is given by Table 6. The general configuration of the board at time t = n is given by Table.: We calculate: P (I) = 4 ( k + k + k + k )=4 4 k = k = (k+) P (E) =P (F )=P (G) =P (H) = 4 ( k + k )= 3 k = (k+) P (C) =P (B) =P (A) =P (D) = 4 ( k )= 4 k = 3 (k+)

AARON ZWIEBACH Table. Possible ositions of walker at even time ste t = n A E F D I B H G C We note that the individual robabilities of each square corresond to those in our inductive hyothesis. By summing all the ositions on the boundary, (A, B, C, D), wecancalculatetherobabilityofourwalkerbeingabsorbed by the border: P [Absorbtion] =4 4 k = (k+) Now that we have a well defined function for the robability of absortion at any even-odd time ste air, we can exressly calculate the exected time to absortion. Theorem 4. The exected time until absorbtion for a walker starting at the center of a 5x5 lattice is 9 time stes. Proof. Using our familiar formula for exectation, we know that the exected time to absortion is given by: X E[Time until absorbtion] = n P (Absorbed at time n) n Given that consecutive even-odd airs ma to the same k, wecansimlify: X = [(k)p (absorbed at time k)+(k +)P (absorbed at time k)] k= = X (4k +)P (absorbed at time k) k= = X (4k +) k k= We can break down our above sum into familiar comonents that we know how to calculate: = X [ k +4k k ] k= The first term, P k= k,isgivenby: X X k = ( )k = k= k= =

RANDOM WALKS AND PERCOLATION 9 And the second term, 4 P k= k k,isgivenby: X 4 k k = =4 = ) k= ( Multilying our two revious sums by,yieldsourdesiredvalueforexected time: ( + ) = 9 3. Exected Size of Tree of Valence d In this section we resent the main result from the aer by Hochstadt, Padilla, and Zwiebach titled: Percolation. This result rovided in Section 5 of the aforementioned aer is an exciting result because using robability and combinatorics in conjunction with generating functions we are able to derive a closed form formula for the exected size of a tree with valence d and oen edge robability. This is another great examle of how tools from the reertoire of discrete mathematics can be used to model and gain insights from natural rocesses. Note: For the uroses of this aer we have made minor edits to the original aer. The edits consist of eliminating all references to other sections in the orginal aer. This has been done for the ease of the reader. Proof as resented in the reffered to aer: 3.. Definitions. This section serves as a reference to rigorously define the symbols and terminology used throughout this aer. Definition 3.. V d, is a ercolation tree of valence d with oen edge robability. Definition 3.. C d, is a cluster within the tree V d,. C d, is the size of the cluster C d,. Definition 3.3. d, is a d-ary ercolation tree with oen edge robability. d, the size of the cluster within the tree d, which contains its root vertex. 3.. Trees of Valence d. We note, that a tree of valence d has a useful relation to (d )-ary trees. A tree of valence d is comrised of a root vertex v, where v has degree d, andeachedgeattachesv to an infinite (d )-ary tree. Again, we denote the size of a cluster in a V d, tree as C d,.similarly, the size of a cluster in a d-ary tree is d,. Assuch,theexectedsizeofa cluster in a V d, tree, E[ C d, ], isgivenby (3.) E[ C d, ]=+d E[ d, ].

0 AARON ZWIEBACH 3.3. Number of Clusters of Size n in a d-ary Tree. We define a recursive function t d (n) that oututs the number of ways to generate a cluster of size n in a d-ary tree. Intuitively, the number of ways to generate a cluster of size n in a d-ary tree is the number of ways of generating clusters in each one of the d subtrees such that the sum of the sizes of the clusters in the sub-trees equals n. More recisely, we can write this as X (3.) t d (n) = t d (a )... t d (a d ). 0alea,...,a d s.t. a +...+a d =n We utilize Equation 3. to define a generating function whose coefficient are the number of ways of generating a cluster of size n X F (x) = t d (n) x n. n=0 By inserting our exression for t d (n) into F (x) we get that X X F (x) d = t d (a )x a t d (a )x a t d (a n )x an = = a =0 X ( n=0 a d =0 X 0alea,...,a d s.t. a +...+a d =n X t d (n +) x n. n=0 t d (a )... t d (a d )) x a + +a n In order for the argument inside t d and the exonent of x to agree we must shift our function over by one: X X xf (x) d = t d (n +) x n+ = t d (n) x n = F (x) which imlies: n=0 n= (3.3) xf (x) d F (x)+=0. When d =, we were able to exlicitly solve for F (x) by alying the quadratic formula. Solving for F (x) in Equation 3.3 is more difficult. In the next section we use a relation between F (x) and G(y) to solve for the exected size of a cluster in a d-ary tree. 3.4. Exected Cluster Size of a d-ary Tree. In order to derive a function for the exected size of of the d-ary tree we do not need to exlicitly solve for F (x). Rather we can use roerties of a d-ary tree to derive a generating function for the robability of yielding a cluster of size n, thatis deendent on F (x). Agivenclusterofsizen is comrised of n oen edges (the edges that connect the n vertices) and n (d ) + closed edges. We can see this by considering what haens when we increase our cluster size by one vertex. When we add another vertex to our cluster we convert one closed-edge to an oen-edge. At the same time, by only adding one vertex, we are adding

RANDOM WALKS AND PERCOLATION d closed-edges. This is because each vertex has d otential children, who in this case are imossible to reach because their edges are closed. As such, each time we add a vertex we also add d closed edges. So a cluster of size n "sees" n (d ) + closed edges. (Where the "+" termaccounts for the n =case.) Utilizing this we can define a generating function G(y) where the coefficient of the y n term is the robability of having a cluster of size n in a d-ary tree: G(y) = X n ( ) n (d )+ t d (n) y n n= = (F ( ( )d y) ). By rearranging the terms of our equality we can avoid having to exlicitly derive F (x). Wedosobyrearrangingthetermssothat (3.4) F (x) = G( x ( ) d )+. At this oint we have a relational equation for F (x) and G(y), givenby Equation 3.4. We can insert our new exression for F (x) in terms of G(y) into Equation 3.3 and simlify by letting y = x : ( ) d ( ) d y ( G(y)+)d G(y) =0. Recall that if we take the derivative of a generating function for the robability of generating a cluster of size n and evaluate the derivative at the oint y =, we get the total exected size of our cluster. Thus we are again looking for G 0 (), which is obtained by taking the derivative of G(y) with resect to y and evaluate at the oint y =. Note that since G(y) is the generating function for the robability density function for cluster size, G() = by the Law of Total Probability. We begin by taking the derivative with resect to y: G0 (y)+( ) d (( G(y)+)d +y(d( G(y)+)d G0 (y))) = 0 We evaluate at y =: G0 () + ( ) d (( +)d +(d( +)d G0 ())) = 0 We simlify: G0 () + ( ) d ( +)d + ( ) d d( +)d G 0 (y) =0

AARON ZWIEBACH We can simlify further: ( +)= : G 0 ()(d ( ) d ( )d )=G0 ()( d G 0 () = d ) We assert that this is only true for values of such that G() =, asthe Law of Total Probability must still hold. This modifies our function G 0 () slightly to become: >< if [0, ) (5.5) G 0 d d () = E[ d, ] = if [ >:, ] d 0 otherwise 3.5. Conclusion. In the last section we derived an exression for the exected cluster size in a d-ary tree (Equation 3.5). At this oint we can tie our result back to Equation 3. and state that >< + d if [0, ) (d ) d E[ C d, ]= if [ d >:, ] 0 otherwise Slight simlification allows us to conclude that Theorem 3 In the grah V d, the exected size of the cluster C d, is: + >< if [0, ) d+ d E[ C d, ]= if [ d >:, ] 0 otherwise 4. Conclusion In this aer we have examined two different natural rocesses. The first comrised of a random walker interacting with a series of boundaries, and the second was the rocess of ercolation. There are a number of viable ways by which one could model a natural rocess mathematically. Given the robabilistic nature of these rocesses one natural inclination might have been to design a comutational simulation that abided by the arameters of the rocess and run it enough times to gather statistically significant data. The aforementioned method is articularly useful for more comlicated structures and boundaries (in the case of the random walker) and for three dimensional orous materials (for the case of ercolation.) While emerical analyses rovide very imortant intuition for more comlex scenarios, they lack the same mathematical definiteiveness that is rovided by a combinatorial or robabilistic roof. We chose these aer recisely because their main results are motivated by latter method of modeling and analysis. In the roblem of the random walker, our authors use attern tracking and induction to derive a function

RANDOM WALKS AND PERCOLATION 3 for the robability that the walker is absorbed at a given time ste. They use this result to further rove that an exected time until absortion for the same boundary. In the ercolation aer our authors use the recursive characteristics of trees and generating functions to count clusters of a given size and leverage those tools to derive a closed form function for the exected size of a valence tree. We invite the reader to exlore this toic further. The two aers: An Analysis of Random Walks in Z and Percolation are attached for the reader to ursue the toic in further detail. The reader will find not only the same method of analysis alied to different shaes of boundaries and instances of valence trees, but also further exloration into the method of emirical analysis, a toic not exlored in deth in this aer. Hay Learning! Massachusetts Institute of Technology, Deartment of Mathematics, Cambridge, Massachusetts, United States E-mail address: aaronz@mit.edu