On the Markov transition kernels for first-passage percolation on the ladder

0.34 MB·English

by  Eckhard Schlemm

ON THE MARKOV TRANSITION KERNELS FOR FIRST-PASSAGE PERCOLATION ON
THE LADDER
ECKHARDSCHLEMM
ABSTRACT. We consider the first-passage percolation problem on the random graph with vertex set N×
{0,1},edgesjoiningverticesatEuclideandistanceequaltounityandindependentexponentialedgeweights.
Weprovideacentrallimittheoremforthefirst-passagetimesln betweenthevertices(0,0)and(n,0),thus
extendingearlierresultsaboutthealmostsureconvergenceofln/nasn→∞. Weusegeneratingfunction
techniquestocomputethen-steptransitionkernelsofacloselyrelatedMarkovchainwhichcanbeusedto
calculateexplicitlytheasymptoticvarianceinthecentrallimittheorem.
2
1
0
2
1. INTRODUCTION
n
a The subject of first-passage percolation, introduced in Hammersley and Welsh (1965) in 1965, is the
J studyofshortestpathsinrandomgraphs. LetG=(V,E)beagraphwithvertexsetV andunorientededges

1 E ⊂V2 and assume that there is a weight function w:E →R . For vertices u,v∈V, a path joining u
+
2
and v in G is a sequence of vertices ppp ={u= p ,p ,...,p ,p =v} such that (p ,p )∈E for
u→v 0 1 n−1 n ν ν+1
0≤ν <n. The weight w(ppp ) of such a path is defined as the sum of the weights of the comprising
] u→v
R edges,w(pppu→v)(cid:66)∑nν−=10w((pν,pν+1)). Thefirst-passagetimebetweentheverticesuandvisdenotedby
P d (u,v)anddefinedasd (u,v)(cid:66)inf{w(ppp),pppapathjoininguandvinG}.
G G
. First-passagepercolationcanbeconsideredamodelforthespreadofafluidthrougharandomporous
h
t medium;itdiffersfromordinarypercolationtheoryinthatitputsspecialemphasisonthedynamicalaspect
a
of how long it takes for certain points in the medium to be reached by the fluid. Important applications
m
includethespreadofinfectiousdiseases(Altmann(1993))andtheanalysisofelectricalnetworks(Grim-
[
mettandKesten,1984). Recently,therehasalsobeenanincreasedinterestinfirst-passagepercolationon

1 graphs where not only the edge-weights, but the edge-structure itself is random. These models, includ-
v ing the Gilbert and Erdo˝s-Re´nyi random graph, were investigated in Bhamidi et al. (2010); Sood et al.
5
(2005); van der Hofstad et al. (2001) and found to be a useful approximation to the internet as well as
8
telecommunicationnetworks.
4
4 Usually, however, the underlying graph is taken to be Z2 and the edge weights are independent ran-
. dom variables with some common distribution P, see e.g. Smythe and Wierman (1978) and references
1
therein. Interesting mathematical questions arising in this context include asymptotic properties of the
0
2 sets Bt (cid:66) {u ∈ Z2 : dZ2(0,u) ≤t} (Kesten, 1987; Seppa¨la¨inen, 1998), and the limiting behaviour of
1 d ((0,0),(n,0))/n. The latter is known to converge, under weak assumptions on P, to a deterministic
Z2
: constant, called the first-passage percolation rate; the computation of this constant has proved to be a
v
i very difficult problem and has not yet been accomplished even for the simplest choices of P (Graham
X
etal.,1995). AnexceptiontothisisthecasewhentheunderlyinggraphGisessentiallyone-dimensional
r (Flaxmanetal.,2011;Renlund,2010;Schlemm,2009).
a
Inthispaperweconsiderthefirst-passagepercolationproblemontheladderG,aparticularessentially
one-dimensional graph, for which the first-passage percolation rate is known (Renlund, 2010; Schlemm,
2009).Weextendtheexistingresultsaboutthealmostsureconvergenceofd ((0,0),(n,0))/nasn→∞by
G
providingacentrallimittheoremaswellasgivingacompletedescriptionofthen-steptransitionkernels
ofacloselyrelatedMarkovchain. Ourresultscanbeusedtoexplicitlycomputetheasymptoticvariancein
thecentrallimittheorem. Theyarealsothebasisforthequantitativeanalysisofanyotherstatisticrelated
tofirst-passagepercolationinthismodel. Inparticular,knowledgeofthehigherordertransitionkernelsis
the starting point for the computation of the distribution of the rungs which are part of the shortest path.
2010MathematicsSubjectClassification. Primary:60K35,60J05;secondary:33C90.
Keywordsandphrases. centrallimittheorem,first-passagepercolation,generatingfunction,Markovchain,transitionkernel.
1
2 ECKHARDSCHLEMM
(cid:72)0,1(cid:76) (cid:72)1,1(cid:76) (cid:72)2,1(cid:76) (cid:72)Ν(cid:45)1,1(cid:76) YΝ (cid:72)Ν,1(cid:76) (cid:72)n(cid:45)1,1(cid:76) (cid:72)n,1(cid:76)
ZΝ
(cid:72)0,0(cid:76) (cid:72)1,0(cid:76) (cid:72)2,0(cid:76) (cid:72)Ν(cid:45)1,0(cid:76) XΝ (cid:72)Ν,0(cid:76) (cid:72)n(cid:45)1,0(cid:76) (cid:72)n,0(cid:76)
FIGURE 1. The ladder graph Gn. The edge weights Xν,Yν, Zν are independent expo-
nentialrandomvariables.
Theladdermodelisworthstudyingbecauseitisoneoftheveryfewsituationswhereacompleteexplicit
descriptionofthefinite-timebehaviourofthefirst-passagepercolationtimescanbegiven.
Thestructureofthepaperisasfollows: InSection2wedescribethemodelandstateourresults;Sec-
tion3, whichcontainstheproofs, isdividedinthreesubsections. Thefirstisdevotedtothecentrallimit
theorem, the second presents some explicit evaluations of infinite sums which are needed in Section 3.3,
wherethemaintheoremaboutthetransitionkernelsisproven. Weconcludethepaperwithabriefdiscus-
sion.
Weusethenotationδ fortheKroneckerdeltaandΘ aswellasΘ˜ forversionsofthediscretized
p,q p,q p,q
Heavisidestepfunction,precisely
(cid:40) (cid:40) (cid:40)
1 if p=q 1 if p≥q 1 if p≤q
δ (cid:66) , Θ (cid:66) , Θ˜ (cid:66) .
p,q p,q p,q
0 else 0 else 0 else
Thesymbol(k)!standsfork!(k+1)!. WedenotebyRtherealnumbersandbyZtheintegers. Asubscript
+(−)indicatesrestrictiontothepositive(negative)elementsofaset. Thesymbolγ standsfortheEuler-
MascheroniconstantandEdenotesexpectation.
2. FIRST-PASSAGEPERCOLATIONONTHELADDER
In this paper we further investigate a first-passage percolation model which has been considered be-
fore in Renlund (2010) and also in Schlemm (2009). We denote by G the graph with vertex setV =
n n
{0,...,n}×{0,1}andedges
X ={((i,0),(i+1,0)):0≤i<n},
n
Y ={((i,1),(i+1,1)):0≤i<n},
n
Z ={((i,0),(i,1)):0≤i≤n}.
n
The edge weights are independent exponentially distributed random variables which are labelled in the
obvious way X, Y and Z, see also Fig. 1. By time-scaling it is no restriction to assume that the edge
i i i
weights have mean one. We further denote by l the length of the shortest path from (0,0) to (n,0) in
n
the graph G , by l(cid:48) the length of the shortest path from (0,0) to (n,1) and by ∆ the difference between
n n n
the two, ∆ =l(cid:48) −l It has been shown in Schlemm (2009) and also, by a different method, in Renlund
n n n
(2010) that lim l /n almost surely exists and is equal to the constant χ = 3 − J1(2) , where J are
n→∞ n 2 2J2(2) ν
Besselfunctionsofthefirstkind. (SeeDefinition1orAbramowitzandStegun(1992)foracomprehensive
treatment.) Thisconstantiscalledthefirst-passagepercolationrateforourmodel. Themethodemployed
inSchlemm(2009)toobtainthisresultbuiltonFlaxmanetal.(2011)andconsistedinshowingthatthere
existsanergodicR×R3-valuedMarkovchainMMM=(M ) withstationarydistributionπ˜ andafunction
+ n n≥0
f :R×R3 →Rsuchthat
+
(cid:90)
χ =Ef(M )= f(m)π˜(dm).
∞
R×R3+
Explicitly,
M =(∆ ,X ,Y ,Z ) and f :(r,x,y,z)(cid:55)→min{r+y+z,x}.
n n n+1 n+1 n+1
ONTHEMARKOVTRANSITIONKERNELSFORFIRST-PASSAGEPERCOLATIONONTHELADDER 3
Throughoutwewritem=(r,x,y,z)forsomeelementofthestatespaceR×R3. Inordertobetterunder-
+
standthefirst-passagepercolationproblemontheladderitisimportanttoknowthehigher-ordertransition
kernelsK˜n:(cid:0)R×R3(cid:1)×(cid:0)R×R3(cid:1)→R oftheMarkovchainMMM.Theycompletelydeterminethedynamics
+ + +
ofthemodelandaredefinedas
(2.1) K˜n(m(cid:48),m)dm=P(cid:0)M ∈dm|M =m(cid:48)(cid:1), m,m(cid:48)∈R×R3.
n 0 +
ThefirstresultshowsthatitissufficienttocontrolthetransitionkernelsKn:R×R→R oftheMarkov
+
chain∆∆∆=(∆ ) whichareanalogouslydefinedas
n n≥0
(2.2) Kn(r(cid:48),r)dr=P(cid:0)∆ ∈dr|∆ =r(cid:48)(cid:1), r,r(cid:48)∈R.
n 0
ForconveniencewedefineK0(r(cid:48),r)(cid:66)δr(cid:48)(r),theDiracdistribution.
Proposition1. Foranyn≥1,denotebyK˜nthen-steptransitionkernelofMMMdefinedinEq.(2.1). Then
(2.3) K˜n(cid:0)m(cid:48),m(cid:1)=e−(x+y+z)Kn−1(cid:0)min{r(cid:48)+y(cid:48),x(cid:48)+z(cid:48)}−min{r(cid:48)+y(cid:48)+z(cid:48),x(cid:48)},r(cid:1).
Moreover,thestationarydistributionπ˜ ofMMMisgivenby
(2.4) π˜(dm)=e−(x+y+z)d3(x,y,z)π(dr),
where
(2.5) π(dr)= 1 e−32|r|J1(cid:16)2e−12|r|(cid:17)dr
2J (2)
2
isthestationarydistributionof∆∆∆.
Next,westateacentrallimittheoremforfirst-passagepercolationtimesontheladderwhichhasbeen
implicitinSchlemm(2009)andwhichwasthemotivationforthecurrentpaper. InAhlberg(2009);Chat-
terjee and Dey (2009) a central limit theorem has been obtained for first-passage times on fairly general
one-dimensionalgraphsbyadifferentmethod. Thequestionofhowtocomputetheasymptoticvariance
has,however,notbeenaddressedthere. Wedenoteby f¯themeancorrectedfunction f−χ.
Theorem 1. For any integer n≥0, let l denote the first-passage time between (0,0) and (n,0) in the
n
laddergraphG . Thenthereexistsapositiveconstantσ2suchthat
n
(2.6) ln−√nχ →−d N(0,σ2),
n
whereN(0,σ2)isanormallydistributedrandomvariablewithmeanzeroandvarianceσ2and→−d denotes
convergenceindistribution. Moreover,
(cid:90) ∞ (cid:90)
(2.7) σ2= f¯(m)2π˜(dm)+2∑ f¯(m)Pnf¯(m)π˜(dm),
R×R3+ n=1 R×R3+
where
(cid:90)
(2.8) Pnf¯(m)=E(cid:2)f¯(M )|M =m(cid:3)= f¯(m(cid:48))K˜n(m,m(cid:48))dm(cid:48).
n 0
R×R3+
Equation (2.7) shows that in order to evaluate the asymptotic variance of the first-passage times one
mustknowthetransitionkernelsK˜n. Inthenexttheoremwethereforeexplicitlydescribethestructureof
thetransitionkernelKnandthus,byProposition1,ofK˜n.Tostatetheformulasinacompactwaywedefine
thefivefunctions
√ √
z−2J (2 z) 2(z−1)+2J (2 z)
S1(z)= 2 , S2(z)= 0 ,
z z
√
z3(cid:20) Y (2 z) (cid:21)
G(z)=− 5−4γ+2π 2 √ −2logz ,
4 J (2 z)
2
z4
α(z)= √ √ √ √ √ and
4 zJ (2 z)[ zJ (2 z)+(z−1)J (2 z)]
2 0 1
z2 (cid:2) √ √ (cid:3)
H(z)= 3z−3+2πY (2 z)+J (2 z)(5−4γ−2logz) .
0 0
2(1−z)
4 ECKHARDSCHLEMM
The Bessel functions J and Y are defined in Definition 1 and treated comprehensively in Abramowitz
ν ν
andStegun(1992).
Theorem2. ThetransitionkernelsKn,definedinEq.(2.2),satisfyKn(r(cid:48),r)=Kn(−r(cid:48),−r). Forr≥0the
valuesKn(r(cid:48),r)aregivenby
∑n an epr(cid:48)−(q+2)r r(cid:48)≤0
(2.9) (−p1,q)n=(−n01−er1(cid:48)p)−,!q(n+1)r +∑np−=20(−1()pn)r!((cid:48)en−−pp(r−(cid:48)−2r))!−nr +∑np,q=0bnp,qe−pr(cid:48)−(q+2)r 0<r(cid:48)≤r,
(−1)n−1e−(n−1)r(cid:48)−r +∑n−2 (−1)nre−p(r(cid:48)−r)−nr +∑n cn e−pr(cid:48)−(q+2)r r(cid:48)>r
(n−1)! p=0 (p)!(n−p−2)! p,q=0 p,q
wherethecoefficientsan ,bn andcn aredeterminedbytheirgeneratingfunctions:
p,q p,q p,q
i) thegeneratingfunctionsAp,q(z)=∑∞n=1anp,qzn, p,q≥0,aregivenby
(−z)q
(2.10a) A (z)= α(z),
1,q (q)!
2(−z)p−1
(2.10b) A (z)= A (z), p≥2,
p,q (p)! 1,q
S2(z)
(2.10c) A (z)= A (z);
0,q 1,q
1−z
ii) thegeneratingfunctionsBp,q(z)=∑∞n=1bnp,qzn, p,q≥0,aregivenby
(−z)q
(2.11a) B (z)= G(z)−A (z),
1,q (q)! 1,q
2(−z)p−1 (−z)p+q+2 p 2k+1
(2.11b) B (z)= B (z)+ ∑ , p≥2,
p,q (p)! 1,q (p)!(q)! k(k+1)
k=2
S2(z) (−z)q
(2.11c) B (z)= B (z)+ H(z);
0,q 1−z 1,q (q)!
iii) thegeneratingfunctionsCp,q(z)=∑∞n=1cnp,qzn, p,q≥0,aregivenby
(−z)q+2
(2.12a) C (z)=d zq+2+B (z)− ,
0,q q 0,q (q)!
(cid:20) zS2(z) (cid:21) z
(2.12b) C (z)= S1(z)+ A (z)− C (z),
1,q 1,q 0,q
2(1−z) 2
2(−z)p−1
(2.12c) C (z)= C (z), p≥2
p,q (p)! 1,q
andthenumbersdqaredeterminedbytheirgeneratingfunctionD(z)=∑∞q=0dqzqgivenby
1(cid:2)√ √ √ √ (cid:3)
(2.13) D(z)= zJ (2 z)(2γ+logz)−π zY (2 z)−1 .
1 1
z
Bythepropertiesofgeneratingfunctionsthecoefficientsan ,bn andcn aredeterminedbythederiva-
p,q p,q p,q
tivesofA ,B andC evaluatedatzero. Thesederivativesareroutinelycalculatedtoanyorderwith
p,q p,q p,q
thehelpofcomputeralgebrasystemssuchasMathematica. Table1exemplifiesthetheorembyreporting
thevaluesofthecoefficientsan ,bn ,cn inthecasen=4. Usingtheseresultstheexpression(2.7)for
p,q p,q p,q
σ2 canbeevaluatedexplicitlyintermsofcertainintegralsofhypergeometricfunctions;thecomputations,
however,arequiteinvolvedandthefinalresultratherlengthy,sowedecidednottoincludethemhere.
3. PROOFS
3.1. ProofsofProposition1andTheorem1. Inthissectionwepresenttheproofsoftherelationbetween
theMarkovchainsMMMand∆∆∆andofthecentrallimittheorem.
ONTHEMARKOVTRANSITIONKERNELSFORFIRST-PASSAGEPERCOLATIONONTHELADDER 5
(A)valuesofa4p,q (B)valuesofb4p,q (C)valuesofc4p,q
q q q
0 1 2 3 0 1 2 3 0 1 2 3
0 115 −17 1 0 721 −13 1 0 721 −13 5 0
96 36 24 432 12 12 432 12 18
1 11 − 1 1 − 1 −241 17 1 0 −241 17 1 0
36 36 12 144 540 48 36 540 48 36
p
2 − 11 1 − 1 0 3 1 0 0 − 1 1 0 0
108 12 72 16 36 144 36
3 1 1 0 0 1 0 0 0 1 0 0 0
72 144 216 216
4 − 1 0 0 0 0 0 0 0 0 0 0 0
1440
TABLE1. Coefficientsofthefour-steptransitionkernelK4,asgiveninEq.(2.9)
ProofofProposition1. Since
∆ =l(cid:48) −l
n n n
=min{l(cid:48) +Y ,l +X +Z }−min{l(cid:48) +Y +Z ,l +X }
n−1 n n−1 n n n−1 n n n−1 n
=min{∆ +Y ,X +Z }−min{∆ +Y +Z ,X }
n−1 n−1 n−1 n−1 n−1 n+1 n−1 n−1
itfollowsatoncethatM beingequaltosomem(cid:48)=(r(cid:48),x(cid:48),y(cid:48),z(cid:48))implies∆ =min{r(cid:48)+y(cid:48),x(cid:48)+z(cid:48)}−min{r(cid:48)+
0 1
y(cid:48)+z(cid:48),x(cid:48)}andthustheMarkovpropertyof∆∆∆togetherwiththeindependenceoftheedgeweightsimplies
thatforanyintegern>1theconditionalprobabilityP(M ∈dm|M =m(cid:48))isgivenby
n 0
e−x+y+zd3(x,y,z)P(cid:0)∆ ∈dr|∆ =min{r(cid:48)+y(cid:48),x(cid:48)+z(cid:48)}−min{r(cid:48)+y(cid:48)+z(cid:48),x(cid:48)}(cid:1).
n 1
ThehomogeneityoftheMarkovchain∆∆∆thenimpliesEq.(2.9)becauseforn=1weclearlyhave
P(cid:0)M1∈dm|M0=m(cid:48)(cid:1)=e−(x+y+z)δmin{r(cid:48)+y(cid:48),x(cid:48)+z(cid:48)}−min{r(cid:48)+y(cid:48)+z(cid:48),x(cid:48)}(r)dm.
Equation(2.4)aboutthestationarydistributionofMMMisadirectconsequenceofthefactthattheedgeweights
X ,Y ,Z areindependentof∆ andexpression(2.5)wasderivedinSchlemm(2009, Proposition
n+1 n+1 n+1 n
5.5.). (cid:3)
Next,weprovethecentrallimittheoremandtheformulafortheasymptoticvarianceσ2.
ProofofTheorem1. We apply the general result Chen (1999, Theorem 4.3.) for functionals of ergodic
Markovchainsongeneralstatespaces. Wefirstnotethat
(cid:90) 2J (2)−3J (2)+ F ({1,1},{2,2,2};−1)−1
f(m)2π˜(dm)= 1 0 2 3 <∞.
R×R3+ J2(2)
ItthensufficestoprovethattheMarkovchainMMMisuniformlyergodic,whichisequivalenttoshowingthat
theMarkovchain∆∆∆isuniformlyergodic.WeusethedriftcriterionAldousetal.(1997,TheoremB),which
assertsthatifthereisasufficientlystrongdrifttowardsthecentreofthestatespaceofaMarkovchain,it
isuniformlyergodic. UsingtheLyapunovfunctionV(r)=1−e−|r|aswellas
(cid:40)
e−|r| ifr(cid:48)<r<0∨r(cid:48)>r>0
(3.1) K(r(cid:48),r)=
e−|r(cid:48)−2r| else
fortheonesteptransitionkernelof∆∆∆(Schlemm,2009,Proposition5.1)weobtain
ψ(r)(cid:66)P1V(r)=1−Ere−|∆1|
(cid:90) 1(cid:104) (cid:105)
=1− e−|ρ|K(r,ρ)dρ = 3−2e−|r|+e−2|r| .
6
R
Itiseasytocheckthat
1 1
ψ(r)≤V(r)− , |r|≥1 and supψ(r)≤supψ(r)= <∞.
10 2
|r|≤1 r∈R
6 ECKHARDSCHLEMM
Since the interval [−1,1] is compact and has positive invariant measure it is a small set (Nummelin and
Tuominen,1982,Remark2.7.) anditfollowsthat∆∆∆isuniformlyergodic,whichcompletestheproof. (cid:3)
3.2. Summationformulas. Inthissectionwederivesomesummationformulaswhichwewilluseinthe
proofsinSection3.3.Someofthemarewell-known,otherscanbecheckedwithcomputeralgebrasystems
suchasMathematica,afew(formulæ8,9,11and12)seemtobenew.Thesumswillbeevaluatedexplicitly
intermsofBesselandgeneralizedhypergeometricfunctions,whichwenowdefine.
Definition1(Besselfunction). Letλ bearealnumberinR\Z . TheBesselfunctionofthefirstkindof
−
orderλ,denotedbyJ ,isdefinedbytheseriesrepresentation
λ
∞ (−1)k (cid:16)x(cid:17)2k+λ
(3.2) J (x)= ∑ .
λ Γ(k)Γ(k+λ+1) 2
k=0
Foranyintegerν theBesselfunctionofthesecondkindoforderν,denotedbyY ,isdefinedas
ν
J (x)cosλπ−J (x)
(3.3) Y (x)= lim λ −λ .
ν
λ→ν sinλπ
Itiswell-knownthatBesselfunctionssatisfytherecurrenceequationJ (x)= 2(ν−1)J (x)−J (x)
ν x ν−1 ν−2
(WolframResearch, Inc.,2010, Formula03.01.17.0002.01), whichweusewithoutfurthermentioningto
simplifyvariousexpressions.
Definition2(Generalizedhypergeometricfunction). Fornon-negativeintegers p≤qandcomplexnum-
bers aaa=a ,...,a and bbb=b ,...,b , b (cid:60)Z , the generalized hypergeometric function of order (p,q)
1 p 1 q j −
withcoefficientsaaa,bbb,denotedby F (aaa,bbb;·),isdefinedbytheseriesrepresentation
p q
∞ (a ) ···(a ) xk
(3.4) F (aaa,bbb;x)= ∑ 1 k p k ,
p q
(b ) ···(b ) k!
k=0 1 k q k
where(z) denotestherisingfactorialdefinedby(z) =Γ(z+k)/Γ(z).
k k
In particular we will encounter the regularized confluent hypergeometric functions F˜ , which are de-
0 1
finedby F˜ ({},{b};−z)= 1 F ({},{b};−z);inthenextlemmawerelatetheirderivativewithrespect
0 1 Γ(b)0 1
tobtocertainsumsinvolvingtheharmonicnumbersHk(cid:66)∑kn=11/n.
Lemma1. Denoteby F˜ theregularizedversionofthehypergeometricfunction F . Itthenholdsthat
0 1 0 1
i) foreverypositiveintegerν,
(3.5a) ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) =γz1−2ν Jν−1(cid:0)2√z(cid:1)−∑∞ k!(k(+−νz)−k 1)!Hk+ν−1.
b=ν k=0
ii) foreverypositiveintegerν,
ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) =(−1)ν−1γzν+21Jν+1(cid:0)2√z(cid:1)−∑∞ k!((−k+z)kν++ν+11)!Hk
b=−ν k=0
ν zk(ν−k)!
(3.5b) +(−1)ν ∑ .
k!
k=0
Proof. For part i), we differentiate the series representation (3.4) term by term. Using the definition of
the Digamma function (cid:122) as the logarithmic derivative of the Gamma function Γ as well as the relation
(cid:122)(k)=−γ+H ,k∈Z (WolframResearch,Inc.,2010,Formula06.14.27.0003.01),weget
k−1 +
d 1 (cid:12)(cid:12)(cid:12) =−(cid:122)(k+ν) = γ−Hk+ν−1
dbΓ(k+b)(cid:12) Γ(k+ν) (k+ν−1)!
b=ν
andthus
ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) =γ ∑∞ k!(k(+−νz)−k 1)!−∑∞ k!(k(+−νz)−k 1)!Hk+ν−1
b=ν k=0 k=0
whichconcludestheproofofthefirstpartofthelemma. Partii)isshowninacompletelyanalogousway,
using the relation lim (cid:122)(µ)/Γ(µ)=(−1)m+1m! for every non-negative integer m, which follows
µ→−m
ONTHEMARKOVTRANSITIONKERNELSFORFIRST-PASSAGEPERCOLATIONONTHELADDER 7
from the fact Wolfram Research, Inc. (2010, Formula 06.05.04.0004.01) that the Gamma function has a
simplepoleat−mwithresidue(−1)m/m!. (cid:3)
Formula1. √
∞ (−z)k 2J (2 z)−z 1
∑ = 2 =− S1(z).
k!(k+2)! 2z 2
k=1
Proof. Immediatefromthedefinition(3.2)ofBesselfunctions. (cid:3)
Formula2. √
∞ (−z)k 1−z−J (2 z) 1
∑ = 0 =− S2(z).
(k+1)!2 z 2
k=1
Proof. Thisisalsoclearfromthedefinition(3.2)ofBesselfunctions. (cid:3)
Formula3.
S3(z)= ∑∞ (−z)n−q =1−J (cid:0)2√z(cid:1)− √zJ (cid:0)2√z(cid:1).
(n−q−2)!(n−q)2 2 1
n=q+2
Proof. Shiftingtheindexofsummationnbyq+2weobtain
∞ (−z)n+2 ∞ (cid:20) 1 1 (cid:21)
S3(z)=∑ = ∑(−z)n+2 −
n!(n+1)!(n+2)2 (n+1)!(n+2)! (n+2)!2
n=0 n=0
∞ (−z)n+1 ∞ (−z)n
=∑ +z−∑ +1−z
(n)! n!2
n=0 n=0
√ √ √
(cid:0) (cid:1) (cid:0) (cid:1)
=1−J 2 z − zJ 2 z
2 1
bythedefinition(3.2)ofBesselfunctions. (cid:3)
Formula4.
(cid:16) (cid:112) (cid:17)
∞ q−2 (−ζz)q J1 2 ζz
Σ1(ζz)= ∑ ∑ = S3(ζz).
(k)!(q−k−2)!(k+2)2 (cid:112)ζz
q=0k=0
Proof. ByFubini’stheoremwecaninterchangetheorderofsummationandthenshiftthesummationindex
qbyk+2toget
∞ (−ζz)k+2 ∞ (−ζz)q
Σ1(ζz)=∑ ∑ .
(k)!(k+2)2 (q)!
k=0 q=0
Bydefinition(3.2),thesecondsumequalsJ (cid:16)2(cid:112)ζz(cid:17)/(cid:112)ζzandsotheclaimfollowswithformula3. (cid:3)
1
Formula5.
∞ q−2 2(−ζz)q (cid:104) (cid:16) (cid:112) (cid:17) (cid:105)(cid:112) (cid:16) (cid:112) (cid:17)
Σ2(ζz)= ∑ ∑ = 2J 2 ζz −ζz ζzJ 2 ζz .
k!(k+2)!(q−k−2)! 2 1
q=0k=1
Proof. Wecaninterchangetheorderofsummationandshifttheindexqbyk+2toobtain
∞ 2(−ζz)k+2 ∞ (−ζz)q
Σ2(ζz)= ∑ ∑
k!(k+2)! (q)!
k=1 q=0
(cid:16) (cid:112) (cid:17) (cid:16) (cid:112) (cid:17) (cid:112)
The first factor equals 2ζzJ 2 ζz −(ζz)2 and the second factor equals J 2 ζz / ζz, both by
2 1
definition(3.2),andsotheclaimfollows. (cid:3)
Formula6.
(cid:16) (cid:112) (cid:17)
∞ (−ζz)q J1 2 ζz
T1(ζz)= ∑ = .
(q)! (cid:112)ζz
q=0
8 ECKHARDSCHLEMM
Proof. Clearfromthedefinition(3.2). (cid:3)
Formula7.
∞ (−ζz)q+1q (cid:16) (cid:112) (cid:17) (cid:112) (cid:16) (cid:112) (cid:17)
T2(ζz)= ∑ =1−J 2 ζz − ζzJ 2 ζz .
(q+1)!2 0 1
q=0
Proof. Thisfollowsfromthedecomposition q = 1 − 1 anddefinition(3.2). (cid:3)
(q+1)!2 (q)! (q+1)!2
Formula8.
∞ (−z)k+2 k 2l+1
U1(z)=∑ ∑
k!(k+2)! l(l+1)
k=1 l=2
√
3z2 (cid:0) √ (cid:1) z (cid:0) √ (cid:1)
= −z−2−πzY 2 z + J 2 z [4γ−3+2logz]
2 1
4 2
(cid:0) √ (cid:1)(cid:20) 5z (cid:21)
+J 2 z 1+ −2γz−zlogz .
0
2
Proof. Firstwenotethat
k 2l+1 k (cid:20)1 1 (cid:21) 5k+3
(3.6) ∑ = ∑ + =2H − ,
k
l(l+1) l l+1 2(k+1)
l=2 l=2
whereH denotesthekthharmonicnumber. ThefirstcontributiontoU1(z)canthereforebecomputedas
k
1 ∞ (5k+3)(−z)k 1 3 1 ∞ (−z)k 5 ∞ (−z)k
∑ =− − − ∑ + ∑
2 k!(k+2)!(k+1) z 4 z (k)! 2 k!(k+2)!
k=1 k=0 k=0
= 1 (cid:104)4J (cid:0)2√z(cid:1)+10√zJ (cid:0)2√z(cid:1)−3z3/2−4√z(cid:105).
4z3/2 1 2
TheothercontributiontoU1(z)is,withthehelpofLemma1,ii),obtainedas
2∑∞ k!((−k+z)2k)!Hk= z22(cid:20)γzJ2(cid:0)2√z(cid:1)−z−1− ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) (cid:21).
k=1 b=−1
ByWolframResearch,Inc.(2010,Formula07.18.20.0015.01),
ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) = 21(cid:2)πzY2(cid:0)2√z(cid:1)− √zJ1(cid:0)2√z(cid:1)[2+logz]+J0(cid:0)2√z(cid:1)[zlogz−1](cid:3)
b=−1
andtheresultfollowsaftercombiningthelastfourdisplayedequations. (cid:3)
Formula9.
(cid:16) (cid:112) (cid:17)
∞ q−2 (−ζz)q k 2l+1 J1 2 ζz
ϒ1(ζz)= ∑ ∑ ∑ = U1(ζz).
(q−k−2)!k!(k+2)! l(l+1) (cid:112)ζz
q=0k=1 l=2
Proof. Interchangingtheorderofthefirsttwosummationsandshiftingtheindexqbyk+2wefindthat
∞ (−ζz)k+2 ∞ (−ζz)q k 2l+1
ϒ1(ζz)= ∑ ∑ ∑ .
k!(k+2)! (q)! l(l+1)
k=1 q=0 l=2
AsbeforethesuminthemiddleequalsJ (cid:16)2(cid:112)ζz(cid:17)/(cid:112)ζzandsotheclaimfollowswithformula8. (cid:3)
1
Formula10.
∞ (−z)n−q
S4(z)= ∑ =z2 F ({1,1},{2,2,2};−z).
(n−q−2)!(n−q−1)2 2 3
n=q+2
Proof. Aftershiftingtheindexnbyq+2thisfollowsimmediatelyfromthedefinition(3.4)ofthehyper-
geometricfunction. (cid:3)
ONTHEMARKOVTRANSITIONKERNELSFORFIRST-PASSAGEPERCOLATIONONTHELADDER 9
Formula11.
∞ (−z)k+2 k 2l+1
U2(z)=∑ ∑
(k+1)!2 l(l+1)
k=1 l=2
z(cid:2) (cid:0) √ (cid:1) (cid:0) √ (cid:1) (cid:3)
= −5+3z+2πY 2 z −2z F ({1,1},{2,2,2},−z)+J 2 z (5−4γ−2logz) .
0 2 3 0
2
Proof. Theproofisanalogoustoformula8. UsingEq.(3.6)thefirstcontributiontoU2(z)is
1 ∞ (5k+3)(−z)k+2 5−3z 5 ∞ (−z)k ∞ (−z)k
∑ = − ∑ −∑
2 (k+1)!2(k+1) 2z 2z k!2 (k+1)!2(k+1)2
k=1 k=0 k=0
1 (cid:2) (cid:0) √ (cid:1) (cid:3)
(3.7) = 5−3z−5J 2 z −2z F ({1,1},{2,2,2};−z) ,
0 2 3
2z
where we used the definitions (3.2) and (3.4). For the remaining part we first use Lemma 1,i) and the
√ √ √ √
identityJ (2 z)/ z−J (2 z)=J (2 z)tocompute
1 2 0
∞ (−z)k
∑ H
(k+1)!2 k+2
k=1
∞ (−z)k ∞ (−z)k
=∑ H +∑ H
(k+1)!(k+2)! k+2 k!(k+2)! k+2
k=1 k=1
(cid:34) (cid:35)
3 1 1 ∞ (−z)k ∞ (−z)k
=− + − ∑ H −z∑ H
2 z z (k)! k+1 k!(k+2)! k+2
k=0 k=0
=−32+1z −γzJ0(cid:0)2√z(cid:1)+1z(cid:20)ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) −z ddb0F˜1({},{b};−z)(cid:12)(cid:12)(cid:12)(cid:12) (cid:21).
b=2 b=3
Itthenfollowsfromtheexplicitcharacterizationof ddb0F˜1({},{b};−z)(cid:12)(cid:12)b=−ν,ν ∈Z+,giveninWolfram
Research,Inc.(2010,Formula07.18.20.0013.01)that
∞ (−z)k 1 (cid:2)√ (cid:2) (cid:0) √ (cid:1)(cid:3) (cid:0) √ (cid:1) √ (cid:0) √ (cid:1) (cid:3)
∑ H = z 2−3z+πY 2 z −2J 2 z − zJ 2 z [2γ+logz] .
(k+1)!2 k+2 2z3/2 0 1 0
k=1
Usingthisweget
∞ (−z)k ∞ (−z)k ∞ (−z)k ∞ (−z)k
2∑ H =2∑ H −2∑ −2∑
(k+1)!2 k (k+1)!2 k+2 (k+1)!2(k+2) (k+1)!2(k+1)
k=1 k=1 k=1 k=1
1 (cid:2)√ (cid:2) (cid:0) √ (cid:1)(cid:3) (cid:0) √ (cid:1) √ (cid:0) √ (cid:1) (cid:3)
= z 2−3z+πY 2 z −2J 2 z − zJ 2 z [2γ+logz]
2z3/2 0 1 0
√
(cid:20) (cid:21)
1 1 J (2 z)
(3.8) + − + 1 +1− F ({1,1},{2,2,2};−z).
2 z z3/2 2 3
CombiningEqs.(3.7)and(3.8)completestheproof. (cid:3)
Formula12.
(cid:16) (cid:112) (cid:17)
∞ q−1 (−ζz)q k 2l+1 J1 2 ζz
ϒ2(ζz)= ∑ ∑ ∑ =− U2(ζz).
(q−k−1)!(k+1)!2 l(l+1) (ζz)3/2
q=0k=1 l=2
Proof. Interchangingtheorderofthefirsttwosummationsandshiftingtheindexqbyk+1wefindthat
∞ (−ζz)k+2 ∞ (−ζz)q−1 k 2l+1
ϒ2(ζz)= ∑ ∑ ∑ .
(k+1)!2 (q)! l(l+1)
k=1 q=0 l=2
(cid:16) (cid:112) (cid:17)
By definition (3.2), the middle sum is equal to −J 2 ζz /(ζz)3/2 and so the result follows readily
1
fromformula11. (cid:3)
10 ECKHARDSCHLEMM
Formula13.
(cid:104) (cid:16) (cid:112) (cid:17)(cid:105) (cid:16) (cid:112) (cid:17)
∞ q−1 2(−ζz)q 2 ζz−1+J0 2 ζz J1 2 ζz
Σ3(ζz)= ∑ ∑ = .
(q−k−1)!(k+1)!2 (cid:112)ζz
q=0k=1
Proof. Thesameasbeforesoweomitit. (cid:3)
Formula14.
∞ q−1 (−ζz)q (cid:112) (cid:16) (cid:112) (cid:17)
Σ4(ζz)=∑ ∑ =− ζzJ 2 ζz F ({1,1},{2,2,2};−ζz).
(k)!(q−k−1)!(k+1)2 1 2 3
q=0k=0
Proof. UsingFubini’stheoremweget
∞ (−ζz) ∞ (−ζz)q−k
Σ4(ζz)= ∑ k ∑ .
(k)!(k+1)2 (q−k−1)!
k=0 q=k+1
The first factor is equal to F ({1,1},{2,2,2};−ζz) by definition (3.4) and the second factor equals
2 3
−(cid:112)ζzJ (cid:16)2(cid:112)ζz(cid:17)bydefinition(3.2). (cid:3)
1
Havingevaluatedthesesumswenowturntorelationsbetweenthemwhichwewillalsoneedandwhich
are proved by writing out the expressions and straightforward computations. The first one only involves
thefivefunctionsappearinginthestatementofTheorem2.
Lemma2. ForalmosteverycomplexnumberzwithrespecttotheLebesguemeasureonthecomplexplane,
(cid:20) S2(z) 1(cid:21) (cid:20)S1(z) S2(z) 1(cid:21) H(z) 3z2
(3.9) + G(z)+ + + α(z)+ + =0.
2(1−z) z z 1−z z 2 4
Lemma3. ForalmosteverycomplexnumberzwithrespecttotheLebesguemeasureonthecomplexplane,
(cid:20)S1(z)−1(cid:21) 3
(3.10) U1(z)+S3(z)+ G(z)= z2−z−1.
z 4
3.3. ProofofTheorem2. Inthissectionweprovethemainresult,Theorem2. First,however,wespend
a closer look on the coefficients an , bn and cn which are defined implicitly through the generating
p,q p,q p,q
functions(2.10)to(2.12).
Lemma4. Foranyintegers p,q≥0thefollowinghold:
(3.11) a1 =δ δ , b1 =0, c1 =0.
p,q p,1 q,0 p,q p,q
Proof. ThisfollowsfromevaluatingthederivativesofthegeneratingfunctionsA ,B ,C atzero. (cid:3)
p,q p,q p,q
Next we derive some useful relations between the coefficients an , bn and cn . These will be the
p,q p,q p,q
mainingredientinourinductiveproofofTheorem2. Thegeneralstrategyinprovingtheequalityoftwo
sequences(sn)n≥1 and(s˜n)n≥1 willbetocomputetheirgeneratingfunctions∑n≥1snzn and∑n≥1s˜nzn and
toshowthattheycoincideforeveryz. Thevalidityofthisapproachfollowsfromthewell-knownbijection
between sequences of real numbers and generating functions (see e.g. Wilf (2006) for an introductory
treatment.) We will constantly be making use of the convolution property of generating functions. By
thiswemeanthesimplefactthatif(s ) isarealsequencewithgeneratingfunctionS(z)and(t ) is
n n≥1 n n≥1
anothersuchsequencewithgeneratingfunctionT(z),thenthesequenceofpartialsums(∑νn−=11tn−νsν)n≥1
has generating function S(z)T(z). We also encounter generating functions of sequences indexed by q
instead of n. In this case we denote the formal variable by ζ instead of z and sums are understood to be
indexedfromzerotoinfinity.
Lemma 5. For every integer n≥1, q≥0 the coefficients defined by the generating functions given in
Theorem2satisfytherelation
q−2bn q−2cn (cid:34)(−1)n(n−1) n−2 (−1)n (cid:35)
(3.12) ∑ k,q−k−2 −∑ k,q−k−2 =δ −∑ .
k+2 k+2 q,n n!2 (k)!(n−k−2)!(k+2)2
k=0 k=0 k=0