Stochastic ProcessesStochastic Processes

Random Walks

Hamid R. Rabiee

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Overview R di A i t Reading Assignment Chapter 10 of textbook

Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic

Processes, 2nd ed., Academic Press, New York, 1975.Processes, 2nd ed., Academic Press, New York, 1975.

2Stochastic Processes

Antony Gormley's Quantum CloudSculpture in London was Sculpture in London was designed by a computer using a random walk algorithmalgorithm

3Stochastic Processes

Outline B i D fi iti Basic Definitions Mathematical Definition Probability of return First return Probability of Eventual Return Application: Satisfiability Problem Application: Satisfiability Problem

4Stochastic Processes

Basic Definitions I RW d l ti l t k it t In a RW model, a particle takes a unit step up or

down at regular intervals, and represents the position of particle at nth step

nSp p p

Examples: The path traced by a molecule The path of a drunk man Gambler’s wealth

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Mathematical Definition C id f i i d d Consider a sequence of i.i.d. random

variables taking the values +1, -1 with probabilities

1 2, ,..., ,...nx x x,p q1 with probabilities

denotes the partial sum of n RVs :,p q

ns

The RW is symmetric if

1 2 ... ...n ns x x x 0 0S

0 5p q The RW is symmetric if 0.5p q

{ } , 2k n kn

nP s r p q r k n

k

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k

Probability of returnW th ti l t d t i i We say the particle returned to origin

at time t if X(t) = 0. F h i l i i i For the particle to return to origin, it

must do m up moves and m down moves.

Lemma: the Probability of returning to i i t dd ti i 0origin at an odd time is 0

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Probability of return (cont.)

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First return

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First return (Cont.)

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First Return Solution

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First Return Solution (Cont.)

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Probability of Eventual Return

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Prob. of Eventual Return (Cont.)

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Application: Satisfiability Problem P bl D fi iti Problem Definition:

Consider n Boolean variables A clause is combination of subset of variables or negation of

1 2, ,..., nx x xg

variables with disjunction (Suppose in this problem each clause has exact two variables):

We have a formula consist of conjunction of clauses. the i jx x

satisfiability problem is either to determine values for the variables that result in the formula being TRUE, or to determine that the formula is never true

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Application: Satisfiability Problem P b bili ti S l ti Probabilistic Solution:

1. Consider an initial TRUE-False random assignment to each variable.

2. If formula isn’t satisfied, consider a clause that is not satisfied and randomly choose one of the Boolean variables in that clause and change its valueg

3. If this new setting makes the formula TRUE then stop, otherwise repeat step 2. If you have not stopped after repetitions, then declare that the 2 (1 4 .75)n p ,formula cannot be satisfied

( )

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Application: Satisfiability Problem S l ti A l i Solution Analysis: Assuming that there is a satisfiable

assignment of truth values and let A be such assignment of truth values and let A be such an assignment

Let denote the number of the n variables jSwhose values at the jth stage of the algorithm agree with their values in A

Thus and with probability at most

j

1S S Thus and with probability at most ½ and with probability at least ½.

1 1j jS S 1 1j jS S

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Application: Satisfiability Problem S l ti A l i (C t ) Solution Analysis (Cont.): Thus we have a random walk Compute probability of reaching from to0S Compute probability of reaching from to

in at most steps Compute probability that we wrongly declare

0 0S

kS n 2 (1 4 .75)n

Compute probability that we wrongly declare that the formula cannot be satisfied

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