Explicit bounds for the return probability of simple random walk

Transcription

1 Explct bouns for the return probablty of smple ranom walk The runnng hea shoul be the same as the ttle.) Karen Ball Jacob Sterbenz Contact nformaton: Karen Ball IMA Unversty of Mnnesota 4 Ln Hall, 7 Church St. SE Mnneapols, MN ) January 9, 4 Abstract We gve an exact computaton of the secon orer term n the asymptotc expanson of the return probablty, Pn, ), of a smple ranom walk on the -mensonal cubc lattce. We also gve an explct boun on the remaner. In partcular, we show that Pn, ) < ` 4πn / where n M s explctly gven. Key wors an phrases. smple ranom walk, ranom walk, return probablty, asymptotc probablty Introucton We stuy here the asymptotc propertes of the smple ranom walk on a mensonal lattce. That s, the sngle step probablty s gven by the formula: { P, j) =, f j =;, otherwse. Insttute for Mathematcs an ts Applcatons, Unversty of Mnnesota, Mnneapols, Mnnesota, kball@ma.umn.eu. Department of Mathematcs, Prnceton Unversty, Prnceton, New Jersey, sterbenz@math.prnceton.eu.

2 Usng ths notaton, one can wrte the probablty of startng at lattce pont an enng at j after n steps of the ranom walk as the convoluton of P wth tself n tmes: Pn, j) = [P ] n, j). In partcular, the probablty of startng at an returnng to the orgn after n steps s gven by the formula: P n, ) = n)! ) n n! n! k +...+k =n ) n!. k!... k! Ths s easly seen by a countng argument, or by evaluatng the constant term n the seres: [ n [ P ), )]n Θ) = e θ + e θ)]..) By orthogonalty, one can also compute the constant term on the rght han se of.) by ntegraton of the entre expresson over the mensonal torus T. That s, evaluaton of the ntegral: = Pn, ) = π) Θ [,π] ) n cos θ Θ..) By Taylor expanng the ntegran of.), one can show that one has the asymptotc: ) / Pn, ). 4πn Ths result s orgnally ue to Pólya, an an account can be foun, for nstance, n [4]. Recently, C. Rtzmann [3] has gven a local central lmt theorem for the strbuton Pnx, y) n the case of farly general ranom walks wth boune step sze on Z. The metho of proof s to agan Taylor expan ntegrals of the form.) whch arse from the Fourer transform of the strbuton functon. Our purpose here s to gve a smple an explct calculaton of a boun for) the ntegral.), whch nclues bouns for all the constants nvolve. Ths s partal progress on a conjecture of Russ Lyons, that n fact the followng boun hols: ) / Pn, ) for all,n..3) 4πn Ths s not ffcult to show for =,, so we concentrate on the case where 3. We prove here that for each fxe, fn M, form suffcently large, the nequalty n.3) s satsfe. Computer calculaton can be then be use to check.3) for the frst M ) terms for a fxe menson. Inee, we have

3 verfe ths for mensons up to an nclung =6. Ths work s use by Felker an Lyons n [] to prove rgorous lower bouns for the followng thermoynamc lmt. Let G n be the graph nuce by the cube of se length n n the -mensonal hypercubc lattce Z an let τg n ) be the number of spannng trees of G n. Then efne h = lm n n ln [ τg n ) ]. Felker an Lyons estmate h usng the followng formula: h =ln) n= Pn, )/n. Snce the terms of the summaton are all nonnegatve, a proof of nequalty.3 gves a metho for computng a rgorous lower boun for h.thus,ourresults can be use to prove a lower boun up to =6. In ths paper, we wll use the followng strct O notaton: hx) = O M fx)), to mean that hx) M fx). Ths notaton follows obvous rules for atons an multplcatons. We wll show that: Proposton. For all n an n, one has the boun: ) [ / Pn, ) 4πn 8n + 7 ) ] +) 4n + + n ) e n..4) Remark. In partcular, the expanson.4) shows that one has Pn, ) < / 4πn) when n M for an M whch can be rea off from the above formula. Ignorng the exponental error for a moment, we see that we nee to at least take n 7 4 +). Hence, M s of the orer 3.When = 3, one sees that the top term n the asymptotc s guarantee to take over at the pont n 79. It s lkely that ths number can be mprove by at least a factor of by tweakng 3

4 the analyss, an wll ncate how ths can be one as we go along. Of course we also nee to regar the exponental error term n the expanson.4), but t s easly seen that ths falls away quckly. In fact, t suffces to tack on one more unt to get n 7 4 +)+ to cover the exponental error. Ths can be seen as follows. We nee to show that for n 7 4 +)+,wehave: ) [ / 4πn 8n + 7 ) ] +) + + ) e n. 4n n Multplyng through by 4n 4πn ) /, estmatng + / n) by, an usng the assumton that n 7 4 +)+,wegetthattsuffcestoshowthatfor approprate an n, 8n ) / 4πn e n/..5) By takng a ervatve, t s easy to show that for fxe ths s a ecreasng functon of n for n n our range. Therefore, t s enough to prove the nequalty n.5) when n = 3 snce 3 < ) +, whch s the smallest value of n n our range) : 8 5 4π ) / e..6) To prove ths, we take another ervatve an show that the left-han se s ecreasng n when 3. Therefore, t suffces to check that.6) hols when = 3, whch t oes. As we have alreay mentone, for n< ) +, computer calculaton can be use to verfy.3) for a fxe menson. We have one ths for = 3, 4, 5, 6. In our analyss, we o not try to rectly compute the ntegral.), but nstea employ a carefully chosen substtute whch s a one mensonal ntegral that bouns the relevant part of).) whle at the same tme t preserves the frst two terms n the asymptotc expanson of.). It shoul be observe that t s also possble to obtan an equalty n.4) by rectly ealng wth.), but the computatons requre woul become qute a bt more nvolve. The man computaton We start wth the ntegral: ) n I = π) cos θ Θ. Θ [ π,π] The man contrbutons to ths ntegral come from near where the ntegran s equal to. That s, moulo an exponental error, we nee only concern ourselves 4

5 wth the ponts,...,) an π,...,π). In fact, by symmetry, we nee only concern ourselves wth the nterval [ π, π ]. Ths allows us to wrte: I = π) Θ N [ π, π ] ) n cos θ ) ) n Θ +O,.) where N s a neghborhoo of the pont,...,) whch we wll explan n a moment. Rearrangng the frst term on the rght han se of.) above, usng the trgonometrc entty cos θ = sn θ, an a rescalng, we may wrte the ntegral as: I = π Θ N [ π 4, π 4 ] ) n sn θ Θ. Now, performng the substtuton u =snθ,wemaywrte: I = π ) n u N [, u u, ] u where N s the mage of N uner ths change of varable. We now choose N so that N s the ball of raus centere at the orgn. Notce that ths s consstent wth the remaner secon term) on the rght han se of.). The choce of raus of ntegraton s the sngle most mportant factor n obtanng boun constants n Proposton. In partcular, the constant of 7 n the thr orer asymptotcs of.4) can probably be mprove by at least a factor of by takng the raus of ntegraton to be smaller. Ths wll of course ncrease the sze of the exponental error. Now, a smple nucton shows that n the range we conser:. u u Ths allows us to boun: I < S π ) n u u u, u where S enotes the volume of the stanar unt sphere n R. Therefore, moulo an exponental error an some constants, we have reuce the problem of bounng Pn, ) to that of stuyng the one mensonal the ntegral: I 3 = u ) n u u u. It s mportant for us to pont out that, although we o not make t explct here, the asymptotc expanson for I 3 matches the asymptotc expanson of the 5

6 orgnal multmensonal ntegral I n the frst two terms. Therefore, the ntegral I 3 ncorporates the mechansm whch s responsble for the boun.3) at least for n suffcently large),.e. a negatve secon asymptotc, whle at the same tme t elmnates many of the error terms that woul come up by tryng to treat I rectly. Ths s the mportant pont of the analyss here. The bouns we wll prouce are as follows: Lemma. For n> an 3, one has that: ln ) I 3 = e n s s ) s + O 4 s 4 ) s..) Proof of Lemma. The entty.) s a straght forwar consequence of Taylor s theorem wth remaner an the followng mplct change of varables: u = e s. Explctly, one has the formulas: s = ln u ),.3) u = e s )..4) Wth ths change of varables an the help of the Taylor expanson: = + u u + O 3 u 4 ), u [, ], we may wrte: ln ) I 3 = e n s s e u ) s + ) s u + O 3 u 4 ) s. Our task s now to gve a Taylor wth remaner formula for the prouct of the last three functons n the ntegran above. For ths, t s useful to note that by the formula.3), one has the boun s 3 over the range of ntegraton. Ths follows from he estmate: ln ) + ), whch can be prove wth th the help of Taylorsformula wth remaner. Therefore, we have that s 3 4. Ths fact wll be use mplctly n all the strct O notaton that follows. We frst compute the Taylor expansons: 6

7 e s = s + O )s4 ),.5) e s = s + ) s 4 + O 3! ) 3s6 )..6) Next, we substtute.6) nto the change of varable formula.4) to see that we may wrte: u = s ) s + O 3! )s4 ),.7) where the expresson s O 3! )s4 ) comes from an alternatng ecreasng seres that s omnate by s. In partcular, we have that ths expresson les n the nterval [, ]. Ths allows us to use the Taylor expanson: x) = x + O as well as the entty: 6 4 ) )max{, } x ), x [, ],.8) ) s O 3! )s4 ) = O 3 s4 ), to set: x = s + O 3! )s4 ), n.8) an expan thngs out an collect terms together to wrte: u s ) = s + O s 4 )..9) Fnally, we can agan use the entty.7) as well as the fact that u = O s) to wrte: + u + O 3 u 4 ) = + s + O 3 s 4 )..) The esre result now follows from multplyng through the expressons on lnes.5),.9), an.) an then usng some rect computatons to show the entty: 7

8 ) s + O )s4 ) ) s + O s 4 ) + ) s + O 3 s 4 ) Ths completes the proof of formula.). = s + O 4 s 4 ).. Wrappng thngs up To get bouns on each of the ntegrals on the rght han se of.), we use the fact that: an ln ) e n s s s < e n s s s, ln ) as well as the fact that: e n s s O 4 s 4 ) s = O 4 e n ln ) s s + s < e n e n s ) s + s < e n e n s s + ) + s < e n )/ s s + e n A lttle further computaton shows that we now have: e n s s +3 s), ) e n s s + s..) π S I 3 < 4πn )/ 4πn )/ 4n )+ 7+) 4πn )/ ) 4n ) + π S R.H.S.).)..) 8

9 We recall the gamma formulas: e s s l s = Γ l + ), S = π Γ ), an use the followng boun whch for nstance follows easly from the ntegral formula 8.3) on p. 87 of []): Γx) < π) / x x e x x /, x, to compute that: π S R.H.S.).) = 4 π Γ ) ) / n e 3 Γ 4n ) / ) / 4e < e n π n + πn < e n n, n. ) n ) /+ + Γ 4n ) 4, n Substtutng ths last nequalty nto the rght han se of.), an usng the nequalty: < e, to tack on the exponental error from lne.), we obtan the result of Proposton. 3 Acknowlegements We wsh to thank Russ Lyons for suggestng ths problem, for reang early versons, an for sharng hs Mathematca coe wth us. K. Ball s work was supporte by an NSF VIGRE grant to Inana Unversty. J. Sterbenz s supporte by an NSF postoctoral fellowshp. References [] Felker, J. L., Lyons, R. 3). Hgh-precson entropy values for spannng trees n lattces. J. Phys. A. 36, no. 3, [] Olver, F. W. J. 974). Asymptotcs an specal functons. Computer Scence an Apple Mathematcs, Acaemc Press. ) ) 9

10 [3] Rtzmann, C. ). Goo Local Bouns for Smple Ranom Walks. avalable at [4] Sptzer, F. 976). Prncples of ranom walks. Secon eton. Grauate Texts n Mathematcs, Vol. 34. Sprnger-Verlag, New York-Heelberg.