; TeX output 2005.09.01:1355 KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ:9|&html: html:.MTvkcolor push Black color popZVg5PSfile=logo129.eps llx=0 lly=0 urx=99 ury=16 rwi=2880Bs獍zcolor push Black color pop_vN q cmbx12A
NoteontheEnZumerationofDiusion bW=alks
intheFirstOctanZtbyTheirNumb erQ
of
ConZtactswiththeDiagonal.s獒 RXQ ff cmr12Heinrich/Niederhausen u&Department/ofMathematicalSciences Florida/AtlanticUniversity BodCca/Raton,FL33431 dUSA @0html:color push cmyk 0 1 0 0߆T ff cmtt12Niederhausen@math.fau.edu html: color pop)
č vcolor push Black color pop r"V
3
cmbx10AbstractMƍlcolor push Black color pop-̻K`y
3
cmr10Diusion0bw!alkstakestepsinthefourdirectionsN,E,S,andW.Weederiveaclosed
_formforthen!umbMerofdiusionw!alksfromtheorigintosomepoin!t( b>
3
cmmi10n;1n)$#.onthe_diagonalfinkstepsinsidetherstoMctan!t,touchingthediagonalexactlyctimes.6ghtml: html:*N G cmbx121(NotationzandResultsb#XQ cmr12A,diusion,wralkisarandomwalkinthesquarelattice/
msbm10Z2|{Y cmr82,=equallylikelytakingoneofthe four|7unitsteps+g cmmi12N@orSthM=(0;1)t,S outh=(0;,!",
cmsy10 1)'u,E ast=(1;0)t,and|7West=( 1;0)'u.Thewralk8staysintherstoSctant,\8alsocalledprincipalwedge,\8ifitonlyvisitslatticepSoints(`n;m)satisfying0URmn.6WVewillcallsucrhrestricteddiusionwalks1@ cmti12offctant#walksinthispapSer.WVedenotebryD22 cmmi8kc(X0K cmsy8! n;c)8thenumbSerofoctanrtwalksfrom(0;0)")to(n;n)$p0taking2kh#stepsandNconrtactingthediagonalyË=URxexactlyctimes.NotethatthestartandendpSointofsuchwralksparecountedascontactswiththediagonal.8Thewalksmayself-intersect,Hbandtheymarytouch(contact)butnotcrosseitherbSoundaryoftheoctanrt.Synonymsforcontactsarevisits,andpffointsDofadsorption.rFigurehtml:color push cmyk 0 1 0 01 html: color popshorwsapathfrom(0;0)""to(3;3)makingfourvisits. :9color push Black 1G color pop *KE&:9color push Blackhtml:color push gray 0 color pop html:G color pop3ڍ+%:9Q color push Black @c {0 1 ^2 У3B|4&U5E&.6 T ff >qڎ
|5 ! ff> ff ffffB{ ff \㳎|4H ff2y# ! ffΟ2 > ffΟ2 ff Т ff {U|3 o ffΟH ff2y# ! ff> ff2"RA ff ^Ʉ ff e|2 + ffΟ 1! o ff2"RA 1!H ff2#RA 1! ! ffΟ 1!> ff2"RA ff ff 9>|1 ff + ff2"RA o ff2"RAH ff2y# ! ff> ff ff { ff ֫|0 G ffΟ 1! ffΟ 1! + ff2"RA o ff2"RAH ffΟ ! ff> ff ff { ff ֫_&. html: html:W?TVablesg1:@html:color push gray 0 color pop html:AsIdiusionwralkintherstoSctantfrom(0;0)to(3;3)with20stepsand4conrtacts color popl-WVeprorveinthispapSerthatD2kc(X0! n;c)8= dcōE,2n+1E,[ z !