Statistics and Probability Letters 127 (2017) 42–48

Contents lists available at ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Computing an expected hitting time for the 3-urn Ehrenfest model via electric networks Yung-Pin Chen a,⇤ , Isaac H. Goldstein a , Eve D. Lathrop b , Roger B. Nelsen a a

Department of Mathematical Sciences, Lewis & Clark College, 0615 SW Palatine Hill Rd., Portland, OR 97219, United States

b

College of Engineering, Oregon State University, 101 Covell Hall, Corvallis, OR 97331, United States

article

info

Article history: Received 4 June 2016 Received in revised form 11 March 2017 Accepted 14 March 2017 Available online 30 March 2017 MSC: 60C05 60J10 Keywords: Ehrenfest urn model Random walk Electric network Hitting time

abstract We study a three-urn version of the Ehrenfest model. This model can be viewed as a simple random walk on the graph represented by ZM 3 , the M-fold direct product of the cyclic group Z3 of order 3, where M is the total number of balls distributed in the three urns. We build an electric network by placing a unit resistance on each edge of the graph. We then apply a series of circuit analysis techniques, including the series and parallel circuit laws and the Delta-Y transformation, to establish shorted triangular resistor networks. A recurrence relation is derived for the effective resistance between two corner vertices of the triangular resistor networks. The recurrence relation is then used to obtain an explicit formula for the expected hitting time between two extreme states where all balls reside in one of the three urns. © 2017 Elsevier B.V. All rights reserved.

1. Introduction We extend the two-urn Ehrenfest model by introducing a third urn. In this three-urn Ehrenfest model, M balls are distributed among urns 0, 1, and 2. At time t = 1, 2, 3 . . . , a ball is chosen at random, removed from the urn it resides in, and placed in one of the other two urns equally likely. As described in Bingham (1991), the process can be modeled as a random walk on a graph at two different levels. 1.1. Two model descriptions In the ‘‘full description’’ we use M-tuples Zt = (Z1,t , Z2,t , . . . , ZM ,t ) of ternary numbers to track the location of each ball at time t. That is, for i = 1, 2, . . . , M, Zi,t assumes a value of 0, 1, or 2 indicating that the ith ball is in urn 0, 1, or 2, respectively, at time t. Because each ball is chosen at random and placed in one of the other two urns with an equal probability, the process {Zt } can be viewed as a simple random walk on the graph represented by ZM 3 , the M-fold direct product of the cyclic group Z3 of order 3. The graph has 3M vertices and 3M M edges in total. Each vertex v is adjacent to 2M vertices whose M-tuple differs from v only by a single component. On the other hand, in the ‘‘reduced description’’ we use vectors (Xt , Yt ) to keep track of the numbers of balls in urn 1 and urn 2, respectively, at time t. In this case, the vertices of the graph representing the three-urn Ehrenfest model are the

⇤

Corresponding author. E-mail address: [email protected] (Y.-P. Chen).

http://dx.doi.org/10.1016/j.spl.2017.03.013 0167-7152/© 2017 Elsevier B.V. All rights reserved.

Y.-P. Chen et al. / Statistics and Probability Letters 127 (2017) 42–48

43

integer lattice points (x, y) in the triangular grid bounded by three constraints: 0  x  M; 0  y  M; 0  x + y  M. The process {(Xt , Yt )} is a random walk on those lattice points, and in fact, a Markov chain with the set of the lattice points as its state space. 1.2. Hitting times and electric networks In both model descriptions, we refer to the graph vertices as states. For a random walk on a graph, the hitting time (or first passage time) from state a to state b is the minimum number of steps the random walk takes to reach vertex b for the first time when the random walk initially starts at vertex a. The expected value of such a hitting time is denoted by Ea Tb . The expected commute time between states a and b is Ea Tb + Eb Ta . The expected hitting times associated with the two-urn Ehrenfest chain are well known. Blom (1989) expresses the expected hitting time between two adjacent states as a definite integral based on a recurrence relation. Bingham (1991) focuses on fluctuation theory and uses generating functions to compute the expected hitting time between the two extreme states. Palacios (1993, 1994) derives closed-form formulas for the expected hitting times via electric networks. One of the key ideas employed by Palacios through electric networks is based on a result due to Chandra et al. (1989), which is stated below. Lemma 1 (Chandra et al., 1989). Let G = (V , E ) be a simple connected graph. Let N (G) be the electric network built by placing a node at each vertex in V and a one-ohm resistor on each edge in E. For each pair of vertices a and b in V , let Ra,b denote the effective resistance between the two given nodes in N (G). Then for the simple random walk on the graph G = (V , E ), the expected commute time from vertex a to vertex b is Ea Tb + Eb Ta = 2 |E | Ra,b , where |E | is the cardinality of the edge set E. 1.3. Our main result and approach The expected hitting times associated with a multiple-urn Ehrenfest model do not appear to be in the literature. In this presentation, we will prove the following main result for the three-urn Ehrenfest model. Main result. For the three-urn Ehrenfest model with M balls, M M balls’’ to the state ‘‘urn 1 has all M balls’’ is M 2M X 3k

3

k=1

k

1, the expected hitting time from the state ‘‘urn 0 has all

.

The hitting time in our investigation involves three extreme states: All M balls are in urn 0, urn 1, or urn 2. We label the three extreme states by a0 , a1 , and a2 , respectively. Our objective is to compute Ea0 Ta1 using the electric network approach. In addition to Lemma 1 and the series and parallel circuit laws used in the process of shorting an electric network, we will employ the Delta-Y transformation. The application of Delta-Y transformation to random walks on a graph is not new. Our colleagues in physics have drawn our attention to the Delta-Y transformation used by Wu et al. (2011) on studying a simple random walk on dual Sierpinski gaskets. 2. Computing Ea0 Ta1 Since the three-urn Ehrenfest model can be regarded as a simple random walk on the graph represented by ZM 3 , we can build an electric network by placing a node at each vertex and a one-ohm resistor on each edge and apply Lemma 1 to compute the expected commute time between a0 and a1 as

Ea0 Ta1 + Ea1 Ta0 = 2 |E | Ra0 ,a1 .

Due to symmetry, we have Ea0 Ta1 = Ea1 Ta0 . With the edge number |E | equal to 3M M, the expected hitting time from a0 to a1 is

Ea0 Ta1 = 3M M ⇥ Ra0 ,a1 .

(1)

So it remains to find the effective resistance Ra0 ,a1 between a0 and a1 for computing the expected hitting time Ea0 Ta1 . 2.1. Shorting the full-description electric network For the electric network built on the simple random walk on ZM 3 , we use an idea by Palacios (1994) to apply a unit voltage between a0 and a1 such that the voltage at a0 is 1 and the voltage at a1 is 0. Then all vertices represented by the M-tuples with the same number of 1’s and the same number of 2’s will have the same voltage and can be shorted. There are M M x such M-tuples with x 1’s and y 2’s, and those vertices are shorted into a vertex represented by the integer lattice x y point (x, y) with 0  x, y, x + y  M. This shorting process yields a triangular grid, which is the reduced description of the three-urn Ehrenfest model. We now find the edge resistances between vertex (x, y) and its nearest neighbors in this

Y.-P. Chen et al. / Statistics and Probability Letters 127 (2017) 42–48

44

shorted triangular electric network. As an example, consider the edge resistance r(x,y),(x M

M x

1,y)

between (x, y) and (x

1, y),

for 1  x  M , 0  y  M. First, there are x M-tuples with x 1’s and y 2’s indicating that urn 1 has x balls and urn y 2 has y balls in the full description model. The transition from vertex (x, y) to vertex (x 1, y) in the reduced description model means that one of the x balls in urn 1 moves to urn 0 and the number of balls in urn 2 remains y. There are x distinct M M x such choices. So each of the x M-tuples with x 1’s and y 2’s is connected to xM-tuples with (x 1) 1’s and y 2’s before y M x

M x y

⇥ x one-ohm resistors in parallel between vertex (x, y) and (x 1, y). It follows from the h i M M x . The edge resistances parallel circuit law that the edge resistance r(x,y),(x 1,y) between (x, y) and (x 1, y) is 1/ x x y between vertex (x, y) and other adjacent vertices in the shorted graph can be computed in a similar manner. We summarize shorting. There are totally

the results below.

Theorem 1. Let G = (ZM 3 , E ) be the graph describing the three-urn Ehrenfest model with M balls. If we build an electric network by placing a node at each vertex in ZM 3 and a one-ohm resistor on each edge in E and reduce the electric network by shorting all nodes represented by the M-tuples with the same number of 1’s and the same number of 2’s, then the shorted electric network can be described by the triangular grid consisting of integer lattice points (x, y) satisfying 0  x, y, x + y  M, and the edge resistances on the edges connecting (x, y) and its nearest neighbors are given as follows. r(x,y),(x

1,y)

r(x,y),(x+1,y) r(x,y),(x,y

1)

 ✓ ◆✓ M M = x  = (M

x

x



M

1,y+1)

r(x,y),(x+1,y

1)

1

M

x

y

x

x

 ✓ ◆✓ M M = y x

M

y

◆

◆

x y

y

◆

1

,

if 0  x  M

1, 0  y  M ;

if 0  x  M , 1  y  M ;

, M

x

x

x

1

✓ ◆✓

y)

 ✓ ◆✓ M M = x

◆

if 1  x  M , 0  y  M ;

,

✓ ◆✓ M M y)

x

r(x,y),(x,y+1) = (M r(x,y),(x

y

◆ x

 ✓ ◆✓

= y

x

x y

1

◆

1

,

if 0  x  M , 0  y  M

,

if 1  x  M , 0  y  M

,

if 0  x  M

1;

1;

1

1, 1  y  M .

The three vertices of an upright-standing triangular circuit can be labeled as (x, y), (x + 1, y), and (x, y + 1). Theorem 1

h

says that both edge resistances r(x,y),(x+1,y) and r(x,y),(x,y+1) are equal to (M

x

y)

on the edge joining (x, y + 1) and (x + 1, y), we may apply the formula r(x,y),(x+1,y r(x,y+1),(x+1,y)

 ✓ ◆✓ M M = (y + 1) x

x

y+1

◆

1

 = (M

x

✓ ◆✓ M M y) x

y

M x y

i

1

. To find the edge resistance

1) with y substituted by y + 1. It gives us

◆

x

M x

1

.

Thus, the shorting process yields an electric network consisting of triangular resistor circuits such that the three edges of an upright-standing triangular circuit have an equal edge resistance. We demonstrate the edge resistances of the shorted triangular electric network for the case of M = 4 in Fig. 1. 2.2. Shorting the reduced-description electric network: Delta-Y transformation Each of the three extreme states a0 , a1 , and a2 is a corner vertex of the shorted triangular resistor network. To find the effective resistance Ra0 ,a1 between a0 and a1 , we employ a circuit analysis technique called the Delta-Y transformation, which is briefly explained below for the sake of completeness. Consider a triangular electric circuit with vertices A, B, and C and edge resistances a, b, and c on the three edges as shown on the left side of Fig. 2. To compute the effective resistance RA,B between vertex A and vertex B, we regard the resistors on edges AC and BC to be in series, and this in-series portion is in parallel with the resistor on edge AB. Applying the series and 1 parallel circuit laws, we obtain that the effective resistance between A and B is RA,B = 1/ a+ + 1c . This -shape triangular b resistor circuit can be transformed to a Y-shape (in fact an upside down Y-shape) resistor circuit with each corner vertex of the -shape circuit as a tip vertex of the Y-shape circuit. Let p, q, and r be the edge resistances on the three edges of the Y-shape circuit shown on the right side of Fig. 2. The effective resistance RA,B between vertex A and vertex B on this Y-shape circuit is p + q. We can find the effective resistances RB,C and RC ,A on both circuits using the same argument. If the -shape circuit and the Y-shape circuit are equivalent, the following equations hold. 1 1 1 RA,B = 1 = p + q; RB,C = 1 = q + r; RC ,A = 1 = r + p. 1 1 1 a +b

+

c

b+c

+

a

c +a

+

b

Y.-P. Chen et al. / Statistics and Probability Letters 127 (2017) 42–48

45

Fig. 1. Two equivalent electric networks for the reduced description of the three-urn Ehrenfest model are displayed for the case M = 4. The electric network on the right is for the Delta-Y transformation.

Fig. 2. Delta-Y transformation.

Fig. 3. The Delta-Y transformation in Eq. (2) is applied to each upright-standing triangular circuit (shaded) for the case M = 4. The edge resistance on an edge of a Y-shape circuit becomes 1/3 of the edge resistance on an edge of the corresponding -shape circuit.

Solving the above system of equations gives us the edge resistances p, q, r in terms of edge resistances a, b, c as follows. p=

bc a+b+c

q=

;

ca a+b+c

;

r =

ab a+b+c

(2)

.

We can also represent edge resistances a, b, c in terms of edge resistances p, q, r. This backward conversion, called the Y-Delta transformation, is given below. a=

pq + qr + rp p

;

b=

pq + qr + rp q

;

c=

pq + qr + rp r

.

(3)

If the three edge resistances a, b, c are equal, then we have p = q = r = a/3. Because the edge resistances on the three edges of an upright-standing triangular circuit are equal, the edge resistances on the three edges of a Y-shape circuit are all equal to a third of the edge resistance on an edge of the -shape circuit under the Delta-Y transformation. This Delta-Y transformation yields an electric network consisting of hexagonal circuits connected to the three corner vertices, as demonstrated in Fig. 3 for M = 4. The resistors on the two outside edges on the same side of a hexagonal circuit are arranged in series, and we may use the series circuit law to short each hexagonal circuit. We demonstrate this shorting process in Fig. 4.

Y.-P. Chen et al. / Statistics and Probability Letters 127 (2017) 42–48

46

Fig. 4. The series circuit law is used to short two outside edges (shaded) of each hexagonal circuit for the case M = 4.

Before the Delta-Y transformation, there are (M + 1) vertices joined by M edges on each side of the triangular electric network. Let us examine the bottom side and label the (M + 1) vertices by coordinates (x, 0)(M ) , 0  x  M, where the superscript (M ) indicates that the vertices are associated with the triangular electric network with M edges on each side. After the Delta-Y transformation and the shorting of the outside edges of each hexagonal circuit, the bottom side, excluding the two extreme state vertices a0 and a1 , comes to have M vertices joined by (M 1) edges. We label those M vertices on the bottom side of the shorted graph by coordinates (x, 0)(M 1) , 0  x  M 1. According to the series circuit law and applying Theorem 1, we can compute the edge resistance r(x,0)(M 1) ,(x+1,0)(M 1) on the shorted electric network as follows. r(x,0)(M

1) ,(x+1,0)(M 1)

=

1 3

r(x,0)(M ) ,(x+1,0)(M ) +

 1 = (M

1

✓ ◆✓ M M x) x

3

 1 = ((M

x)

1)

3

✓

3

r(x+1,0)(M ) ,(x+2,0)(M ) x

0

M

◆

1

◆✓

1 x

+

1 3

(M

 (M

1)

(x + 1))

◆

1

x

0

✓

M

◆✓

x+1

M

(x + 1) 0

◆

1

.

Similar computation shows that the same results hold for the other two sides of the shorted triangular electric network. After shorting the outside edges of each hexagonal circuit using the series circuit law, the main body of the electric network, excluding the three corner vertices and the three edges incident to them, turns into a triangular shape again. Inside this triangular electric network, there are solely (upright-standing) Y-shape circuits, as illustrated in the graph on the left of Fig. 5 for M = 4. The center vertex of each Y-shape circuit is in fact an interior node of the original triangular electric network. If we label this center vertex by (x, y)(M ) , where 0 < x, y < M and x + y < M, then the edge resistances on the northwest, northeast, and south edges of the Y-shape circuit are, respectively, p= q= r =

1 3 1 3 1 3

r(x,y)(M ) ,(x

1,y)(M )

=

r(x,y)(M ) ,(x,y+1)(M ) = r(x,y)(M ) ,(x,y

1)(M )

=

1 3 1 3 1 3

 ✓ ◆✓ x

M

M

x

 (M

x

 ✓ ◆✓ y

M

x y

1

M

M

M

x

x y

(4)

,

✓ ◆✓

y)

x

◆

◆

x y

1

◆

1

(5)

,

(6)

.

Now we apply the Y-Delta transformation to each Y-shape circuit. This backward transformation will make the center vertex (x, y)(M ) of each Y-shape circuit vanish, and at the same time, it will create a triangular resistor circuit with vertices (x 1, y)(M 1) , (x, y)(M 1) , (x, y 1)(M 1) , listed clockwise from northwest, to northeast, and to south, in the resulting shorted graph. Using the equations in (3), together with the edge resistances in (4), (5), and (6), we have r(x,y)(M r(x,y

1)(M 1)

=

1)(M 1) ,(x 1,y)(M 1)

=

r(x,y)(M

1) ,(x,y

1) ,(x

1,y)(M 1)

=

pq + qr + rp p

pq + qr + rp q

pq + qr + rp r

= = =

1 3 1 3 1 3

 ✓ y

 ✓ x

 ✓ x

M

◆✓

1 x

M

1 x

M

1 x

◆✓

M

y

M

◆✓

M

x

1

x

1 y

1 1 y

x

◆

◆ ◆

1

, 1

, 1

.

Y.-P. Chen et al. / Statistics and Probability Letters 127 (2017) 42–48

47

Fig. 5. The Y-Delta transformation in Eq. (3) is applied to each Y-shape resistor circuit (shaded) inside the triangular electric network on the left for the case M = 4, and it converts each Y-shape circuit to a triangular circuit displayed on the right.

The computation above confirms that the formulas in Theorem 1, multiplied with a constant factor 1/3, also hold for the edge resistances on an interior edge after the Y-Delta transformation. An illustration of this shorting process for the case M = 4 is given on the right of Fig. 5. 2.3. A recurrence relation for computing Ra0 ,a1 (M )

Now we let Ra0 ,a1 denote the effective resistance between the two extreme states a0 and a1 on the triangular electric network representing the three-urn Ehrenfest model with M balls. For M = 1, the electric network is simply a triangular (1) (M ) resistor circuit. The Delta-Y transformation yields Ra0 ,a1 = 2/3. To compute Ra0 ,a1 for M 2, we can add the edge (M )

(M )

resistance ra0 ,b0 = 1/(3M ) between a0 and b0 , the effective resistance Rb0 ,b1 between b0 and b1 , and the edge resistance (M ) rb1 ,a1 = 1/(3M ) between b1 and a1 , where b0 and b1 are the two corner vertices on the shorted electric network as (M )

(M 1)

demonstrated in the graph on the right of Fig. 5 for M = 4. Our work has shown Rb0 ,b1 = (1/3)Ra0 ,a1 , and it establishes the recurrence relation ) R(aM = 0 ,a 1

2 3M

1

+ R(aM0 ,a11) ,

M

3

2,

(7) (M )

(1)

with the initial condition Ra0 ,a1 = 2/3. Solving this recurrence relation, we can express Ra0 ,a1 as a finite sum: ) R(aM = 0 ,a 1

M 2X

3 k=1

✓ ◆M

k

1

1 k

3

M

,

1.

(8)

So now we are ready to use Eq. (1), where the effective resistance Ra0 ,a1 is expressed without the superscript (M ), to compute Ea0 Ta1 as follows.

Ea0 Ta1 = 3M M ⇥ Ra0 ,a1 = 3M M ⇥

M 2X

3 k=1

✓ ◆M 1

k

1 k

3

=

M 2M X 3k

3

k=1

k

,

M

1.

This completes proving our main result, as claimed in the introduction. 3. Concluding remarks

We conclude our presentation by a conjecture on a generalized n-urn Ehrenfest model, for n 4. In the two-urn Ehrenfest model, there are only two extreme states. Using electric networks, Palacios (1993, 1994) computes the effective resistance between the two extreme states to be

PM

equivalent expression of R(M ) is R(M ) = 2 R(1) = 1,

R(M ) =

1

1

1 1 1 M 1 . Denote this quantity by k=0 M k M 2k M (M ) satisfies the recurrence k=1 k , and R

+ R(M

P

R(M ) . It can be shown that another relation

1) , M 2. (9) M 2 We observe a similarity between this recurrence relation in (9) and that in (7). For the n-urn Ehrenfest model, we conjecture that the recurrence relation for the effective resistance between any two extreme state vertices is

R(1) = 2/n,

R(M ) =

2 nM

1

+ R(M n

1)

,

M

2.

(10)

Y.-P. Chen et al. / Statistics and Probability Letters 127 (2017) 42–48

48

The initial condition R(1) = 2/n can be verified as follows. For the n-urn Ehrenfest model with M = 1 ball, the electric n network can be represented by the complete graph Kn with n vertices and a one-ohm resistor on each of the 2 = n(n 1)/2 edges. It is known (see page 2498 in Ellens et al., 2011) that the total effective resistance of Kn is (n 1). So the effective resistance between any two vertices of Kn is 2/n, which justifies the initial condition. For the 3-urn case, the electric network can be represented by a planar triangular grid with the three corner vertices denoting the three extreme states. However, for the n-urn case with n 4, we will have four or more extreme states, and we need a grid with 3 or higher dimensions to represent the electric network. The process of shorting such a high dimensional electric network becomes more complicated, and it makes the establishment of the recurrence relation in (10) more challenging. If the conjectured recurrence relation in (10) can be shown to be true, we can find the solution to the recurrence relation to be (M )

R

=

M 2X

n k=1

✓ ◆M 1

k

n

1 k

,

M

1.

M With the number of edges of ZM 1)/2, we conjecture the expected hitting time between two extreme n equal to n M (n states of the n-urn Ehrenfest model with M balls to be M

n M

(n

1) 2

⇥

M 2X

n k=1

✓ ◆M 1 n

k

1 k

=

(n

M 1)M X nk

n

k=1

k

.

Acknowledgments We thank the Miller Foundation, the Howard Hughes Medical Institute, Shannon O’Leary’s Lab, the Community Engagement and Leadership in Science program and the Rogers Foundation of Lewis & Clark College for their support. References Bingham, N.H., 1991. Fluctuation theory for the Ehrenfest urn. Adv. Appl. Probab. 23, 598–611. Blom, G., 1989. Mean transition times for the Ehrenfest urn model. Adv. Appl. Probab. 21, 479–480. Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R., Tiwari, P., 1989. The electrical resistance of a graph captures its commute and cover times. In: Proceedings of The 21st Annual ACM Symposium On Theory of Computing, Seattle, Washington, pp. 547–586. Ellens, W., Spieksma, F.M., Van Mieghemc, P., Jamakovicb, A., Kooijb, R.E., 2011. Effective graph resistance. Linear Algebra Appl. 435, 2491–2506. Palacios, J., 1993. Fluctuation theory for the Ehrenfest urn via electric networks. Adv. Appl. Probab. 25, 472–476. Palacios, J., 1994. Another look at the Ehrenfest urn via electric networks. Adv. Appl. Probab. 26, 820–824. Wu, S., Zhang, Z., Chen, G., 2011. Random walks on dual Sierpinski gaskets. Eur. Phys. J. B 82, 91–96.