Transcription of INTRODUCTION TO STOCHASTIC PROCESSES. MARKOV …
1 6 February 2017 INTRODUCTION TO STOCHASTIC RANDOM FIELDSD avid A. MeyerDepartment of MathematicsUniversity of California/San DiegoLa Jolla, CA @dajmeyerRecall that in doing Problem 1 on the first midterm we learned that the weather todaydepends upon the weather yesterday and tomorrow, even assuming the MARKOV s make a short excursion to generalize this idea, and in so doing, answer the questionof what we might mean by a MARKOV property for a STOCHASTIC process whose index set issomething other than (a subset of) the AgraphG= (V, E) consists of a setVofverticesand a symmetric relationE V V. Elements ofEare callededges; each is an unordered pair of For anyv V, let theneighborhoodofv,N(v) ={w V|(v, w) E}, ,all vertices inVthat are connected tovby an Given a graphG= (V, E), let{Xv|v V}be a STOCHASTIC process (withindex setV). Suppose that for allv V,Pr(Xv=xv Xw=xw, w V\{v})= Pr(Xv=xv Xw=xw, w N(v)).Then the STOCHASTIC process is aMarkov random field.
2 We understand this as sayingthat the random variableXv, conditioned on the values of the random variables at theneighboring vertices, is independent of the remaining random consider a very simple graph withV=Zn, the integers modulon N, so we canlabel themV={0,1, .. , n 1}), andEconsisting of all the pairs (i, i+ 1) where + isinterpreted modulon. In other words,Gis a ring ofnvertices. Define a STOCHASTIC process{Xi { 1,+1} i Zn}, with joint probability function forX= (X0, .. , Xn 1) isPr(X=x) =1Z( , n)e ixixi+1,c David A. Meyer 20171 STOCHASTIC processes : MARKOV random fieldsDavid A. Meyerwhere >0 and thepartition function,Z( , n) = xe ixixi+1.( )This is a one-dimensionalIsing modelwith periodic boundary conditions. It is a model formagnetic materials:{ 1,+1}denote spin down and spin up for a magnetic moment,and is (proportional to) the inverse Prove that this STOCHASTIC process is a MARKOV need to show that, for example, Pr(X0=x0|X1=x1, X2=x2.
3 , Xn 1=xn 1) = Pr(X0=x0|X1=x1, Xn 1=xn 1). By the definition of conditionalprobability:Pr(X0=x0|X1=x1, X2=x2, .. , Xn 1=xn 1)=Z 1( , n)e (x0x1+x1x2+ +xn 1x0)Z 1( , n)e (x0x1+x1x2+ +xn 1x0)+Z 1( , n)e ( x0x1+x1x2+ xn 1x0)=e (x0x1+xn 1x0)e (x0x1+xn 1x0)+e (x0x1+xn 1x0),which is independent of the variablesx2, .. , xn ComputeZ( , n), , simplify ( ).LetTuv=e uv, whereu, v { 1,+1}. Then thetransfer matrixisT=( 1 +1 1e e +1e e ).We can write the partition function as the trace of the product ofncopies ofT:Z( , n) = TrTn,to compute which we need to find the eigenvalues ofT. Solving0 = det(e e e e )= 2 2e +e 2 e2 ,gives =e e , soZ( , n) = 2n(coshn + ( 1)nsinhn ).c. What is limn Z( , n)?For fixed >0, cosh >sinh , so asn ,Z( , n) 2ncoshn .c David A. Meyer 20172
