Random Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary


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1 Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes sraegies for specific desired oucomes A random walk is basically looking a he ime sequence in a binomial process. We will only consider he simples random walks here
2 Random Walk in D Consider an asymmeric random walk along he ais, beginning a he origin. There is a probabiliy p ha he sep will be +, and qp ha he sep will be. We could addiionally define he probabiliy o say pu. 5 3 possible pahs vs n 5 5 For our random walk, we assume he probabiliies p,q do no depend on ime n  saionary
3 Random Walk in D D Random Walk: Firs quesion: wha is he probabiliy o be a afer n seps? Le n + represen he number of seps in he + direcion and n  he number of seps in he  direcion. Then n + n n + +n n so n + n+/ n n/ Noe ha his is only possible if n+ is even!
4 Random Walk in D The disribuion for n + is a Binomial disribuion: so <n + >np Now for Pn + n p n + q nn + n! n + n +!nn +! pn + q nn + <n + >npq+n p σ n+ <><n + n><n + >nnp npq < ><n + n >4 <n + >4<n + >n+n 4npq+n 4pq σ 4npq
5 Random Walk Eample: symmeric random walk pq/. Then σ npq 4 n Package We now look a he probabiliy of reurning o he origin. Use he symbol P,n o represen he probabiliy of being a afer n seps. P,n n! n +!n! pn+ q n
6 Take n even and. Random Walk Change variables o simplify he mah somewha: m n P,m m! m!m! pm q m For large n, use Sirling s approimaion: m! m m+/ e m π P,m mm+/ e m m m+ e m π pm q m m pq m mπ 4 pqm mπ
7 Random Walk P,m 4 pqm mπ Noe ha pq /4, so ha P,m for m. The probabiliy o be a he origin goes o. However, he number of reurns o he origin afer N seps Taking pq/, RN N/ P,m m N/ m! RN / m N +! m m!m! N N N! N/ m! p m q m m m!m! N π Sirling s appro. The sae is said o be recurren  here is probabiliy one of evenually reurning o he origin. Only rue of pq/.
8 Random Walk in D Le fm be he probabiliy ha m is ever reached m>. Then: f m + f m f The probabiliy ha we ge o m+ a some ime is jus he probabiliy ha we go o m a some earlier ime imes he probabiliy ha we ever ge o + from here. This implies: f m f m Wha is f? Saring a he origin, we can eiher reach m on he firs sep, or we have o each i saring from . I.e., f p + qf p + qf
9 We can solve for f: Random Walk in D f ± 4 pq q ± 4 p p p ± p p For p /, we ake he negaive sign: f p p p q For p>/, we ake he negaive sign f The chance o reach m a some ime is for p>/, m>, and p/q m for p</, m>.
10 Random walk in D Now le gm,n be he probabiliy of reaching he poin +m before n. How do we find his quaniy very imporan if have absorbing boundaries. Sar wih wha we know: f m gm,n + [ gm,n] f m + n Prob o evenually ge o m firs reached n firs Go here by firs reaching m f m gm,n + [ gm,n] f m+n Solving for gm,n gives This is for p</. p gm,n q m p q m+n p q m+n
11 Random Walk in D Eample: A physicis wih some special powers is able o guess correcly he flip of a coin wih 6% probabiliy. He sars wih wo marbles, and plays a game of guessing he oss wih someone who has an infinie number of marbles. Calculae he probabiliy ha he physicis will ulimaely lose boh marbles. Our formula works for p</, so we ake p and calculae he probabiliy o ge o before reaching  g, f f f since f I.e., beer han 5% chance of winning all he marbles!
12 Random Walk in D Anoher eample: suppose you have Euros and play roulee. You be on black every ime, so have 8/38 chance of being correc on any given spin. Wha is he probabiliy of making Euros before losing? Wha is he bes sraegy? Suppose you make Euro bes: p.474, q.56, p/q.9, m, n gm,n A beer sraegy would be o make large bes! Try 5 Euro bes. gm,n
13 Random Walk Fun fac: he symmeric random walk in 3 dimensions has a nonzero probabiliy of never reurning o he saring poin.
14 Random Walk in D Now a somewha rickier problem: Wha is he probabiliy ha he walk reurns o he origin a he n h sep for he firs ime? Clearly, we need n o be even. Le us firs look a he case where he firs sep is o he righ So now we have a random walk saring a and ending up a and here are n seps. n n sep
15 The probabiliy for ending a afer n seps is n! P,n n +! n! pn+ q n n!! n +! n We need o subrac all pahs which reach in our shifed Coordinae sysem.. To coun hese, we use a symmery argumen. Noe ha once we are a, hen we have equal probabiliy o end up a  and +. Therefore, he probabiliy o hi and end up a + is he same as he probabiliy o end up a  saring from +. I.e P reach , n P, n n The probabiliy no o reach in our n seps is herefore P, n P, n
16 We now pu he pieces ogeher. We have wo possibiliies: sar lef or sar righ: P F [ P, n P, ], n P, n / n firs ime o reurn Le s ry i for a few cases n [ ] / P, n / 4 F By consrucion, we see ½ he pahs reurn o zero for he firs ime a n n F [ 3/ 8 / 4] / 6 6 P,6 / 4 By consrucion, we see 4 pahs reurn o zero for he firs ime a n6, from a oal of / 6 pahs, or 4/64/6.
17 Brownian Moion We consider a physics eample of a random walk Brownian moion. Discovered in 87 by he English boanis Brown, who observed ha small paricles immersed in a liquid ehibi irregular moion. Mahemaical descripion from he laws of physics by Einsein in 95, who sared wih he assumpion ha he moion was caused by repeaed collisions of he molecules wih he medium. Subjec of inense ineres since. hp://
18 Brownian Moion Formulaion: Le X denoe he displacemen from he saring poin projeced ino a fied ais a ime. The displacemen X X over he ime inerval  can be views as a large number of small displacemens. We posulae ha he disribuion of X X does no depend on he values of, bu only on he inerval . We can hen apply he cenral limi heorem and epec ha X X follows a Gaussian disribuion. Le be he componen a ime, I.e., X. The probabiliy of finding he paricle a a disance a a laer ime + is p,. The normalizaion condiion is saisfied:  And we assume ha p, d lim p,
19 Brownian Moioncon. Einsein showed ha: p D p The condiional probabiliy for he posiion of he paricle follows a diffusion equaion, wih diffusion coefficien D given by D RT N f Where R is he gas consan, T he emperaure, N Avogadro s number and f he coefficien of fricion. By judicious choice of unis, we can se D/. The soluion o he diffusion equaion is hen p, e π
20 Brownian Moioncon. Check Gaussian is a soluion o he diffusion equaion: 3 so we ge, Facoring ou p p e p e p e e e p e p e e p π π π π π π π π
21 Brownian Moion con. Anoher approach sar from symmeric random walk: To ge o posiion a sep n+, we have o be a eiher  or + a sep n: P,n + P, n + P +, n Now rewrie by subracing P,n from each side P,n + P,n [ P +, n P,n + P, n ] Noice ha his looks like a firs derivaive in ime on he LHS of he equaion  remember ha n is a ime variable  and a nd derivaive of space of he RHS n fied. So, we recover he diffusion equaion from he random walk. Need physics inpu o ge he unis.
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