Green functions and Martin compactification for killed random walks related to SU(3)

We consider the random walks killed at the boundary of the quarter plane, with homogeneous non-zero jump probabilities to the eight nearest neighbors and drift zero in the interior, and which admit a positive harmonic polynomial of degree three. For these processes, we find the asymptotic of the Green functions along all infinite paths of states, and from this we deduce that the Martin compactification is the one-point compactification.


Introduction and main results
First introduced for Brownian motion by R. Martin in 1941, the concept of Martin compactification has then been extended for countable discrete time Markov chains by J. Doob and G. Hunt at the end of the fifties.The purpose of this theory is to describe the asymptotic behavior of the Markov chains and also to characterize all their non-negative superharmonic and harmonic functions, see e.g.[Dyn69].
For a transient Markov chain with state space E, the Martin compactification of E is the smallest compactification Ê of E for which the Martin kernels z → k z 0 z = G z 0 z /G z 1 z extend continuously -by G z 0 z we mean the Green functions of the process, i.e. the mean number of visits made by the process at z starting at z 0 , and we note z 1 a reference state.Ê \ E is usually called the full Martin boundary.Clearly, for α ∈ Ê, z 0 → k z 0 α is superharmonic ; then ∂ m E = {α ∈ Ê \ E : z 0 → k z 0 α is minimal harmonic} is called the minimal Martin boundary -a harmonic function h is said minimal if 0 ≤ h ≤ h with h harmonic implies h = ch for some constant c.Then, every superharmonic (resp.harmonic) function h can be written as h(z 0 ) = Ê k z 0 z µ(dz) (resp.h(z 0 ) = ∂mE k z 0 z µ(dz)), where µ is some finite measure, uniquely characterized in the second case above.
The case of the homogeneous random walks in Z d is now completely understood.First, their minimal Martin boundary is found in [DSW60], thanks to Choquet-Deny theory.Furthermore, in the case of a non-zero drift, P. Ney and F. Spitzer find, in their wellknown paper [NS66], the asymptotic of the Green functions, by using exponential changes of measure and the local limit theorem ; this gives consequently the concrete realization of the Martin compactification, in that case the sphere.Additionally, in the case of a drift zero, the asymptotic of the Green functions is computed in [Spi64] ; it follows that the Martin compactification consists in the one-point compactification.
Results on Martin boundary for non-homogeneous random walks are scarcer and more recent.We concentrate here our analysis on important and recently extensively studied examples that are the random walks in Z d killed at the boundary of cones.They are related to many areas of probability, as e.g. to non-colliding random walks or quantum processes.
On the one hand, the case of the non-zero drift is now rather well studied.In [Bia92b], P. Biane considers quantum random walks on the dual of compact Lie groups and, by restriction, arrives at classical random walks with non-zero drift killed at the boundary of the Weyl chamber of the dual.Solving an equation of Choquet-Deny type, he finds the minimal Martin boundary of these processes.
When the compact Lie group is SU(d) and the associated random walk has non-zero drift, the Martin compactification is obtained in [Col04], by finding the asymptotic of the Green kernels.
Recently, in [IR09], I. Ignatiouk-Robert obtains the Martin compactification of the random walks in Z d + with non-zero drift and killed at the boundary.She uses there an original approach based on large deviations theory in order to compute the asymptotic of the Martin kernels.Unfortunately, her methods seem quite difficult to extend up to the case of the drift zero.Also, they do not provide the asymptotic of the Green functions.
This asymptotic in the case of the dimension d = 2 is found in [KR09].
On the other hand, results on Martin boundary for killed random walks with drift zero are quite rare.The simplest example of the cartesian product is due to [PW92].A more interesting case comes again from quantum processes : in [Bia92a], P. Biane shows that the minimal Martin boundary of the random walk with zero drift and killed at the boundary of the Weyl chamber of the dual of SU(d) is reduced to one point.
By the same methods, it can certainly be shown that there is only one positive harmonic function for the "dual" walk, namely for the random walk with jump probabilities p −1,0 = p 0,1 = p 1,−1 = 1/3.In particular, if we set P c = {random walks of P such that in other words, P c is the set of all cartesian products of the random walk on the dual of SU(3) with its dual, see on the left of Picture 2 below -, it follows from [PW92] that any process of P c has also a minimal Martin boundary reduced to one point.
In this paper, we introduce the set P 1,0 = {random walks of P for which (i 0 , j 0 ) → i 0 j 0 (i 0 + j 0 ) is harmonic}.
Note that we have P c ⊂ P 1,0 , but we will see, in Remark 4, that the inclusion is strict.More generally, we define P α,β = {random walks of P for which (i 0 , j 0 ) → i 0 j 0 (i Since any harmonic function for a killed process takes the value zero on the boundary, P α,β is in fact exactly the set of all killed random walks in Z 2 + to the eight nearest neighbors admitting a harmonic polynomial of degree three. The description of the set P α,β in terms of the (p i,j ) i,j is rather cumbersome but not difficult to obtain, it is postponed until Remark 4. Let us just note here that if α > 2 or α < 1/2, then for all β, P α,β = ∅ ; if α = 1/2 or α = 2, then for all β = 0, P α,β = ∅, and P α,0 is reduced to one walk ; and if α ∈]1/2, 2[ and |β| is small enough, then P α,β is a (non-empty) set with two free parameters, properly described in Remark 4. We have represented on the right of Picture 2 an example of a process belonging to P α,0 , for any Moreover, note that considering in this paper P α,β is all the more natural as the set {random walks of P for which (i 0 , j 0 ) → i 0 j 0 is harmonic} is studied in [Ras09].
Our first result deals with the Green functions -below, (X, Y ) denotes the coordinates of the random walk and E (i 0 ,j 0 ) the conditional expectation given (X(0), Y (0)) = (i 0 , j 0 ) - and, more precisely, with their asymptotic along all paths of states.
Theorem 1. Suppose that the process belongs to P α,β .Then the Green functions (2) admit the following asymptotic as i + j → ∞ and j/i → tan(γ), γ lying in [0, π/2] : where C > 0 depends only on the parameters (p i,j ) i,j and is made explicit in the proof.
In the particular case of the random walk killed at the boundary of the Weyl chamber of the dual of SU(3), the asymptotic (3) is, for γ ∈]0, π/2[, proved in [Bia91].Theorem 1 actually completes this result for that very particular random walk and, in fact, gives the asymptotic of the Green functions for a much larger class of processes.
In addition, Theorem 1 has the following consequence.
Corollary 2. The Martin compactification of any process belonging to P α,β is the onepoint compactification.
Furthermore, the asymptotic (3) of the Green functions in the two limit cases γ = 0 and γ = π/2 enables us to obtain the asymptotic of the absorption probabilities Indeed, the absorption probabilities (4) are related to the Green functions (2) through so that, from Theorem 1, we immediately obtain the following result.
The asymptotic of the absorption probabilities in the case of a non-zero drift being obtained in [KR09], Corollary 3 thus gives an example of the behavior of these probabilities in the case of a drift zero.
In order to prove Theorem 1, we are going to develop methods initiated in [FIM99] and based on complex analysis, what will allow us to express explicitly the Green functions (2).Indeed, in [FIM99], the authors G. Fayolle, R. Iasnogorodski and V. Malyshev elaborate a profound and ingenious analytic approach for studying the stationary probabilities for random walks to the eight nearest neighbors in the quarter plane supposed ergodic, i.e.
We are going to see here that this analytical approach can be extended up to the case of the random walks in the quarter plane with drift zero and killed at the boundary : Section 2 of this paper first broadens the analysis begun in Part 6 of [FIM99] for the drift zero, and then shows how this applies in the case of the random walks of P α,β .
It is worth noting that this approach via complex analysis is intrinsic to the dimension d = 2 ; for this reason, it seems really a difficult task to generalize it in higher dimension.
Let us conclude this introductory part by describing the set P α,β defined in (1) in terms of the jump probabilities (p i,j ) i,j .
Remark 4. The fact that the two drifts are equal to zero gives two equations and the fact that the sum of the jump probabilities is one yields an other one.Moreover, the harmonicity of h(i 0 , j 0 ) = i 0 j 0 (i 0 + αj 0 + β), which reads h(i 0 , j 0 ) = i,j p i,j h(i 0 + i, j 0 + j), leads to ten equations, by identification of the coefficients of the third-degree polynomials above.It turns out that some of these equations are trivial and that some other ones are linearly dependent, we finally obtain six equations linearly independent.We can therefore express all the eight jump probabilities (p i,j ) i,j in terms of p 1,1 and p 1,0 only, and we obtain :

Explicit expression of the Green functions
Section 2 aims at obtaining an explicit expression of the Green functions ( 2) -what we will succeed in doing in Theorem 8 below.This forthcoming expression of the Green functions will be, in turn, the starting point of Section 3, where we will find their asymptotic.
In order to prove Theorem 8, we need to state two results, namely Equation ( 6) and Proposition 6 : Equation ( 6) is a functional equation between the generating function of the Green functions (2) and the ones of the absorption probabilities (4), and Proposition 6 establishes some quite important properties of the generating functions of the absorption probabilities.
The proof of Proposition 6 turns out to require considering the Riemann surface defined by {(x, y) ∈ (C ∪ {∞}) 2 : i,j p i,j x i y j = 1}, for this reason we begin Section 2 by studying -and, in fact, by uniformizing -this surface.
It seems of interest to us to introduce this Riemann surface in whole generality ; this is why, at the beginning of Section 2, we are going to suppose that the process belongs to P -and not necessarily to P α,β .
To begin with, we define the generating functions of the Green functions (2) and of the absorption probabilities (4) by and Of course, G i 0 ,j 0 , h i 0 ,j 0 and hi 0 ,j 0 are holomorphic in their unit disc.With these notations, we can state the following functional equation on where Q(x, y) = xy i,j p i,j x i y j −1 .Equation ( 6) is obtained exactly as in Subsection 2.1 of [KR09].
The polynomial Q(x, y) defined above can obviously be written as Let us also define the polynomials It is proved in Part 2.3 of [FIM99] that for any random walk of P, d (resp.d) has one simple root in ] − 1, 1[, that we call x 1 (resp.y 1 ), a double root at 1, and a simple root in R ∪ {∞} \ [−1, 1], that we note x 4 (resp.y 4 ).
From a general point of view, it is shown in Part 2.3 of [FIM99] that x 1 (resp.y 1 ) is positive, zero or negative depending on whether p −1,0 ) is positive, zero or negative, and that x 4 (resp.y 4 ) is positive, infinite or negative depending on whether p 1,0 ) is positive, zero or negative.
Let us now have a look to the surface defined by {(x, y) As a consequence, it follows from the particular form of d or of d (two distinct simple roots different from 1 and one double root at 1) that the surface Q has genus zero, and is thus homeomorphic to a sphere C ∪ {∞}, see e.g.Parts 4.17 and 5.12 of [JS87].Therefore, this Riemann surface can be rationally uniformized, in the sense that it is possible to find two rational functions x(z) and y(z), such that Q = {(x(z), y(z)) : z ∈ C ∪ {∞}} ; moreover, a standard uniformization (for an account of the concept of uniformization, see Part 4.9 of [JS87]) is : where and where K is a complex number of modulus 1.Note that z 0 and z 1 (resp.z 2 and z 3 ) have a modulus equal to one or are real, according to the signs of x 1 and x 4 (resp.y 1 and y 4 ).For example, in the case of SU(3), it follows from a direct calculation that Above and throughout the paper, we note ı the usual complex number verifying ı 2 = −1.
In the general case, in order to find K, we need to introduce a group of automorphisms naturally associated with the surface Q.To begin with, let us remark that, with the previous notations, Q(x, y) = 0 entails Q(x, [c(x)/a(x)]/y) = 0 and Q([c(y)/ã(y)]/x, y) = 0 ; it is therefore natural to consider the group generated by the two bilinear transformations ξ(x, y) = (x, [c(x)/a(x)]/y) and η(x, y) = ([c(y)/ã(y)]/x, y), which is called, in [FIM99], the group of the random walk.
These automorphisms ξ and η define two automorphisms ξ and η of the uniformization space C ∪ {∞}, characterized by : With ( 7) and (8), we obtain that they are equal to : In particular, it is immediate that the group W = ξ, η generated by ξ and η is isomorphic to the dihedral group of order 2 inf{n > 0 : K 2n = 1}.For example, in the case of SU(3) for which K = exp(−ıπ/3), W is of order six -this fact is (differently) proved in Part 4.1 of [FIM99].
A crucial fact is that this property is actually verified by any random walk of P α,β , since we have the following.
From now on, we suppose that the process belongs to P α,β .
For a better understanding of the surface Q as well as for a coming use, we are now going to be interested in the transformations through the uniformization (x, y) of some important cycles, namely the branch cuts [x 1 , x 4 ], [y 1 , y 4 ] and the unit circles {|x| = 1}, {|y| = 1}.First, by using (7) and Proposition 5, we immediately obtain : As for the cycles x −1 ({|x| = 1}) and y −1 ({|y| = 1}), their explicit expression (calculated starting from (7)) shows that they are real elliptic curves, which are located as in the middle of Picture 3 below.Note also that with (9) and Proposition 5, we immediately obtain ξ(exp(ıθ)R + ) = exp(−ıθ)R + and η(exp(ıθ)R + ) = exp(−ı(θ + 2π/3))R + .In particular, if we denote by F the set {x exp(ıθ) : x ≥ 0, −π/3 ≤ θ ≤ 0}, we have -see also on the right of Picture 3 - Thanks to the group W = ξ, η and to (11), we are now going to continue the lifted functions H i 0 ,j 0 (z) = h i 0 ,j 0 (x(z)) and Hi 0 ,j 0 (z) = hi 0 ,j 0 (y(z)) ; this fact will turn out to be of the highest importance in the proof of Theorem 8 -the latter being crucial, since it will be the starting point of the forthcoming Section 3.
Note that in the sequel, we are often going to write x i 0 y j 0 (z) instead of x(z) i 0 y(z) j 0 .
Proof of Proposition 6.In order to prove Proposition 6, we are going to use strongly the decomposition (11) : precisely, we are going to define H i 0 ,j 0 and Hi 0 ,j 0 piecewise, by defining them on each of the six domains w(F ) that appear in the decomposition (11), to be equal to some functions H i 0 ,j 0 w and Hi 0 ,j 0 w .It will then be enough to show that the functions H i 0 ,j 0 and Hi 0 ,j 0 so defined verify the conclusions of Proposition 6.
Picture 1, we are going to use the most natural way to define H i 0 ,j 0 and Hi 0 ,j 0 , i.e. their power series.So we set, for z ∈ F , H i 0 ,j 0 1 (z) = h i 0 ,j 0 (x(z)) and Hi 0 ,j 0 1 (z) = hi 0 ,j 0 (y(z)) -the subscript 1 standing for the identity element of the group W = ξ, η .

Proof of Theorem 1 : asymptotic of the Green functions
Beginning of the proof of Theorem 1.For any θ ∈ [2π/3, π], the function x(z) i y(z) j is, on exp(ıθ)[0, ∞], larger than 1 in modulus, see Picture 3.Moreover, it goes to 1 when (and only when) z goes to 0 or to ∞.This is why it seems natural to decompose the contour exp(ıθ)[0, ∞] into a part near 0, an other near ∞ and the remaining part, and to think that the parts near 0 and ∞ will lead to the asymptotic of G i 0 ,j 0 i,j , and that the remaining part will lead to a negligible contribution.In this way appears the question of finding the best possible contour in order to achieve this idea ; in other words, it is a matter of finding the value of θ for which the calculation of the asymptotic of (14) on exp(ıθ)[0, ∞] will be the easiest, among all the possibilities θ ∈ [2π/3, π].
For this, we are going to consider with details the function x(z) i y(z) j , or, equivalently, the function χ j/i (z) = ln(x(z)) + (j/i) ln(y(z)).Incidentally, this is why, from now on, we suppose that j/i ∈ [0, M ], for some M < ∞.Indeed, the function χ j/i is manifestly not appropriate to the values j/i going to ∞, for such j/i, we will consider later the function (i/j)χ j/i (z) = (i/j) ln(x(z)) + ln(y(z)).Nevertheless, M can be so large as wished, and, in what follows, we assume that some M > 0 is fixed.Now we set χ j/i (z) = p≥0 ν p (j/i)z p , this function is a priori well defined for z in a neighborhood of 0.Moreover, with (7), we obtain that ν 0 (j/i) = 0 and that for all p ≥ 1, Likewise, we easily prove, by using (7), that for z near ∞, χ j/i (z) = p≥0 ν p (j/i)z −p .Consider now the steepest descent path associated with χ j/i , that is the function z j/i (t) defined by χ j/i (z j/i (t)) = t.By inverting the latter equality, we immediately obtain that the half-line (1/ν 1 (j/i))[0, ∞] is tangent at 0 and at ∞ to the steepest descent path.Now we set, for the sake of briefness, ρ j/i = 1/ν 1 (j/i).With this notation, let us now answer the question asked above, that dealt with finding the value of θ for which the asymptotic of G i 0 ,j 0 i,j will be the most easily calculated : we are going to choose θ = arg(ρ j/i ) ∈ [2π/3, π], and the decomposition of the contour exp(ıθ)[0, ∞] will be By using this decomposition in (14), we consider now that the Green functions are the sum of three terms, and we are going to study successively the contribution of each of these terms.
Equation ( 21) implies then that the integral (14) on the contour (ρ j/i /|ρ j/i |)[0, ǫ] equals So, with (18) and (20) applied for k = 2 and k = 5, we obtain that the integral (14) on the contour (ρ j/i /|ρ j/i |)[0, ǫ] is equal to Contribution of the neighborhood of ∞.The part of the contour close to ∞, Therefore, the change of variable z → 1/z immediately entails that the contribution of the integral (14) near ∞ is the complex conjugate of its contribution near 0.
Let us now consider the function s(z) = 1 x i 0 y j 0 (z) w∈W (−1) l(w) x i 0 y j 0 (w(z)) , and let us show that sup z∈D |s(z)| is finite.For this, it is sufficient to prove that s has no pole in the closed domain D ∪ {∞}.By (7), the only zeros of the denominator of s are at z 1 , 1/z 1 , Kz 3 , K/z 3 which, as we easily check, belong to −(D ∪ D).Also, by ( 7) and (9), the only poles of the numerator of s are at K 2k z 0 , K 2k /z 0 , K 2k+1 z 2 , K 2k+1 /z 2 , for k ∈ {0, 1, 2}.Next, we verify that both z 0 and Kz 2 belong to D, so that among the twelve previous points, in fact only z 0 and Kz 2 are in D. But in the definition of s, we took care of dividing by x i 0 y j 0 , so that s is in fact holomorphic near these two points.Moreover, s is clearly holomorphic at ∞. Finally, we have proved that the meromorphic function s has no pole in the closed domain D ∪ {∞}, hence s is bounded in D ∪ {∞}, in other words sup z∈D |s(z)| is finite.
In particular, the modulus of the contribution of the integral (14) on the intermediate part (ρ j/i /|ρ j/i |)]ǫ, 1/ǫ[⊂ D ∩ A ǫ can be bounded from above by Note that the presence of the term 1/ǫ 2 in ( 23) is due to the following : one 1/ǫ appears as an upper bound of the length of the contour, the other 1/ǫ comes from an upper bound of the modulus of the term 1/z present in the integrand of (14).Then, as before, we take ǫ = 1/i 3/4 , and we use the following straightforward upper bound, valid for i large enough : 1/(1 + η/i 3/4 ) i ≤ exp(−(η/2)i 1/4 ).We finally obtain that for i large enough, (23) is equal to O(i 3/2 exp(−(η/2)i 1/4 )).We are going to see soon that this contribution is negligible w.r.t. the sum of the contributions of the integral (14) in the neighborhoods of 0 and ∞.
In order to prove Theorem 1 in the case γ = π/2, we would consider (i/j)κ j/i rather than κ j/i , and we would use then exactly the same analysis, we omit the details.
A few words about the analytical approach used here.The two key steps in the proof of Theorem 1 are first the explicit expression for the Green functions (15), and then the expansion (13b) of H i 0 ,j 0 + Hi 0 ,j 0 + h i 0 ,j 0 0,0 − x i 0 y j 0 at 0, which is the numerator of the integrand in (15).
It is worth noting that for any walk of P ⊃ P α,β , it is still possible to obtain (15) -without additional technical details, besides.On the other hand, obtaining explicitly the expansion at 0 of H i 0 ,j 0 + Hi 0 ,j 0 + h i 0 ,j 0 0,0 − x i 0 y j 0 in the general setting seems us quite difficult -all the more so as this expansion has to comprise several terms, since a priori it could happen that several terms lead to non-negligible contributions in the asymptotic of the Green functions.
It is more imaginable (though technically difficult) to obtain this expansion for the walks for which an equality like (13b) holds ; unfortunately, having such an equality is far for being systematic, even for the processes associated with a finite group W : for example, the random walk with jump probabilities p 1,1 = p 0,−1 = p −1,0 = 1/3 has manifestly a group W of order six, but does not verify an identity like (13b).

Figure 1 :
Figure 1: Random walk in the Weyl chamber of the dual of SU(3)

FFigure 3 :
Figure 3: The uniformization space C ∪ {∞}, with on the left some important elements of it, in the middle the corresponding elements through the uniformization (x, y), and on the right the images of the cone F = {x exp(ıθ) : x ≥ 0, −π/3 ≤ θ ≤ 0} through the six elements of the group W = ξ, η