Long time behavior of random walks on the integer lattice

We consider an irreducible finite range random walk on the d-dimensional integer lattice and study asymptotic behavior of its transition function p(n; x) close to the boundary of Cramér’s zone.


Communicated by A. Constantin.
B Bartosz Trojan btrojan@impan.pl 1 Instytut Matematyczny Polskiej Akademii Nauk, ul.Śniadeckich 8, 00-656 Warsaw, Poland uniformly with respect to n ∈ N and x ∈ Z d . As it may be easily seen, the local limit theorem is very inaccurate when |x| 2 is larger than √ n. Further development of the Fourier method allowed to gain better control over the error term for large |x| 2 (see Smith [12], Spitzer [13, P7.10], Ney and Spitzer [10,Theorem 2.1]). Namely, uniformly with respect to x ∈ Z d \{0}. Let us observe that the error in the approximation of p(n; x) is additive and may become large compared to the first term. In many applications, it is desired to have an asymptotic formula for p(n; x) valid on the largest possible region with respect to n and x. There are some results in this direction available. In particular, (see Lawler [ Let us emphasize that the above asymptotic formula is valid for random walks having exponential moment. However, applied to the simple random walk gives the asymptotic formula useful only in the region where |x| 2 = o(n 3/4 ). Therefore, there arises a natural question: Is there an asymptotic formula for p(n; x) which is valid on a larger region than |x| 2 = o(n 3/4 )?
The purpose of this article is to give a positive answer to the possed question. To be more precise, let us first introduce some notation. Let p be the transition function of the irreducible finite range random walk. By V we denote its support, namely Let M be the interior of the convex hull of V. For δ ∈ M we set where κ is a function on R d defined by the formula We also need a quadratic form on R d given by B x (u, u) = D 2 u log κ(x). In Sect. 3 we prove the following theorem.
Theorem A Let p be a transition function of an irreducible finite range random walk on Z d . Let r be its period and X 0 , . . . , X r −1 the partition of Z d into aperiodic classes. There is η ≥ 1 such that for each j ∈ {0, . . . , r − 1}, n ∈ N and x ∈ X j , if n ≡ j (mod r ) then otherwise p(n; x) = 0, where δ = x n and s = ∇φ(δ).
Some comments are in order. First, observe that the asymptotic formula (2) is valid in a region excluding only the case when n dist(δ, M) 2η stays bounded. Although the function φ is positive convex and comparable to | · | 2 2 , see Claim 1, it cannot be replaced in the asymptotic formula by | · | 2 2 without introducing an additional error term, see Remark 1. For processes with continuous time it was observed by Davis in [3] that in order to get an upper bound for the heat kernel on a larger region one has to introduce a non-Gaussian factor. Therefore, Theorem A may be considered as a discrete time counterpart of [3]. Finally, although the quadratic form B x is explicitly given, the mapping M δ → s(δ) is an implicit function. We want to stress the fact that while δ approaches the boundary of M, the value of |s| 2 tends to infinity. In particular, the quadratic form B s degenerates. For this reason a more convenient form of Theorem A is given in Corollary 3.2.
A positive answer to the posed question is partially given in [2, Theorem 4.1], however thanks to Theorem A we get the control over the error term.
Let us comment about the method of the proof of Theorem A. First, with the help of the Fourier inversion formula, we write p(n; x) as an oscillatory integral. We split the integral into two parts. The first part we analyze by the Laplace method. This is not a straightforward application of it, since the phase function degenerates as δ approaches the boundary of M. To estimate the second part we develop a geometric argument (see Theorem 2.2), which allows us to control the way how the quadratic form B s degenerates. In fact, Theorem 2.2, is the main observation of the present paper. It can be successfully applied in much wider context. For example, to study finitely supported isotropic random walks on affine buildings (see [14]). The result obtained in this article has already found an application in the study of subordinated random walks (see [1]) which are spread all over Z d and do not have second moment. There is also an ongoing project to get the precise asymptotic formula for random walks with internal degrees of freedom extending the one obtained by Krámli and Szász [7] (see also Guivarc'h [4]). Finally, Appendix A contains applications of Theorem 3.1 to triangular and hexagonal lattices. This has to be compared with results recently obtained in [5,6].

Notation
We use the convention that C stands for a generic positive constant whose value can change from line to line. The set of positive integers is denoted by N. Let N 0 = N∪{0}.

Random walks
Let p(·, ·) denote the transition density of a random walk on the d-dimensional integer lattice. Let p(x) = p(0, x). For n ∈ N 0 and x ∈ Z d we set The support of p is denoted by V, i.e.
We assume that the set V is finite. Let κ : C d → C be an exponential polynomial defined by where · , · is the standard scalar product on C d , i.e., In particular, R d θ → κ(iθ) is the characteristic function of p. We set Finally, the interior of the convex hull of V in R d is denoted by M.
In this article, we study the asymptotic behavior of transition functions of irreducible finite range random walks. Let us recall that the random walk is irreducible if for each x ∈ Z d there is n ∈ N such that p(n; x) > 0. By r ∈ N we denote the period of p, that is r = gcd n ∈ N : p(n; 0) > 0 .
For each x ∈ Z d , by m x we denote the smallest m ∈ N such that p(m; x) > 0, thus x/m x ∈ M. Notice that there is C ≥ 1 such that for all x ∈ Z d , Indeed, let {e 1 , . . . , e d } be the standard basis of R d . Since by setting ε j = sign x, e j we get Hence, which, together with boundedness of M, implies (4). Next, we observe that there is K > 0 such that for all k ≥ K , p(kr; 0) > 0, thus for all x ∈ Z d and n ≥ K r + m x , Since do not lay on the same affine hyperplane, the interior of the convex hull of (6) is a non-empty subset of M.
For each x ∈ R d , by B x we denote a quadratic form on R d defined by where D u denotes the derivative along a vector u, i.e.
In particular, if the random walk is irreducible then the quadratic form B x is positive definite.
Example 1 Let p be the transition function of the simple random walk on Z d , i.e.
By a straightforward computation we may find the quadratic form B 0 ,

Function s
For the sake of completeness we provide the proof of the following well-known theorem.
attains its maximum at the unique point s ∈ R d satisfying ∇ log κ(s) = δ.
Proof Without loss of generality, we may assume ∇κ(0) = 0. Indeed, otherwise we will considerκ Since ∇κ(0) = 0, by Taylor's theorem we have as |x| 2 approaches zero. Moreover, for any x, u ∈ R d , Since M is not empty, the set V cannot be contained in an affine hyperplane, thus 0 ∈ M.
and we get and the proof is finished.
In the rest of the article, given δ ∈ M by s we denote the unique solution to Let φ : M → R be defined by thus, by Theorem 2.1, By (9), for any u ∈ R d , Hence, for u, u ∈ R d , i.e. D u s = B −1 s u. Therefore, we can compute In particular, φ is a convex function on M.
as δ approaches δ 0 . We claim that Since φ is convex and satisfies (12), it is enough to show that φ is bounded from above. Given δ ∈ M, let v 0 ∈ V be any vector satisfying proving the claim.
Hence, using (11) we obtain In general, there is no explicit formula for the function φ. By the implicit function theorem, the function s is real analytic on M. In particular, s is bounded on any compact subset of M. On the other hand, |s| 2 approaches infinity when δ tends to ∂M. To see this, let us denote by F a facet of M such that δ approaches ∂M ∩ F . Let u be an outward unit normal vector to M at F . Then for each Therefore, for any v ∈ V\F , The next theorem provides a control over the speed of convergence in (13).

Theorem 2.2
There are constants η ≥ 1 and C > 0 such that for all δ ∈ M, and v ∈ V we have Proof We consider any enumeration of elements of where S is the unit sphere in R d centered at the origin. Since V is finite, it is enough to prove that there are C > 0 and η ≥ 1 such that for all Without loss of generality, we may assume that = ∅. Let k be the smallest index such that points {v 1 , . . . , v k } do not lay on the same facet of M. Let us recall that a set F is a facet of M if there are λ ∈ S and c ∈ R such that for all v ∈ V, λ, v ≤ c, and Since {v 1 , . . . , v k } do not lay on the same facet of M and is a compact set, there is > 0 such that for all ω ∈ we have Indeed, otherwise, there are ω n ∈ such that Since is compact, there is ω 0 ∈ such that and for each i ∈ {2, . . . , N }, This contradicts that {v 1 , . . . , v k } do not lay on the same facet of M. Let F be a facet containing {v 1 , . . . , v k−1 } determined by λ ∈ S and c ∈ R. Let us consider x ∈ R d such that x The distance of δ to a plane containing the facet F is not bigger than c − λ, δ , thus we obtain In particular, for 1 ≤ j ≤ k, we have If j > k, we can estimate which finishes the proof since, by (14), thus it is enough to take Example 3 For k ∈ N, let us consider a transition probability, Then for δ ∈ (−k, 2) we have

Analytic lemmas
For a multi-index σ ∈ N d we denote by X σ a multi-set containing σ (i) copies of i. Let σ be a set of all partitions of X σ . For the convenience of the reader we recall the following lemma.

Lemma 2.3 (Faà di Bruno's formula)
There are positive constants c π , π ∈ σ , such that for sufficiently smooth functions f : Let us observe that for the function F( f (s)) is real-analytic in some neighborhood of s = 0, thus there is C > 0 such that for every σ ∈ N d , where for a multi-set B containing σ (i) copies of i we set

Using Lemma 2.3 one can show
Lemma 2.4 Let V ⊂ R d be a set of finite cardinality. Assume that for each v ∈ V, we are given a v ∈ C, and b v > 0. Then for z = x + iθ ∈ C d such that Moreover, there is C > 0 such that for all σ ∈ N d , Proof We start by proving (16). We have For the proof of (17), it is enough to show where in the last inequality we have used (15).

Heat kernels
In this section we show the asymptotic behavior of the n'th step transition density of an irreducible finite range random walk on the integer lattice Z d . Before we state and prove the main theorem, let us present the following example.

Example 4
Let p be the transition function of the simple random walk on Z. If x ≡ n (mod 2) then Let us recall Stirling's formula Hence, we have Theorem 3.1 Let p be an irreducible finite range random walk on Z d . Let r be its period and X 0 , . . . , X r −1 the partition of Z d into aperiodic classes. There is η ≥ 1 such that for each j ∈ {0, 1, . . . , r − 1}, n ∈ N and x ∈ X j , if n ≡ j (mod r ) then otherwise p(n; x) = 0, where δ = x n , s = ∇φ(δ) and Proof Using the Fourier inversion formula we can write where Hence, by (5), which implies that e it is r 'th root of unity. In particular, the set U has the cardinality r . Next, we claim that To see this, we observe that if y ∈ R d and x = y, then we have We notice that if p(n; x) > 0 then δ = x n ∈ M. Since dist(δ, ∂M) > 0, by Theorem 2.1, there is the unique s = s(δ) such that ∇ log κ(s) = δ. Hence, by Claim 2, we can write Let > 0 be small enough to satisfy (25), (27) and (30). We set Then the integral over D d is negligible. To see this, we write Now, we need the following estimate.

Claim 3 For every
For the proof, we assume to contrary that for some v 0 ∈ V and all m ∈ N there is θ m ∈ D d such that for all v ∈ V, In order to apply Claim 3, we select any v 0 satisfying thus e s,v 0 ≥ κ(s). By Claim 3 and (22), for each θ ∈ D d there is v ∈ V such that Although v may depend on θ , by Theorem 2.2, there are C > 0 and η ≥ 1 such that for all θ ∈ D d , Hence, Since n dist(δ, ∂M) η = n 1 2 n dist(δ, ∂M) 2η 1 2 we obtain that provided n is large enough. We observe that for any u ∈ R d , thus for any u, u ∈ R d , and hence Therefore, we conclude that Next, let us consider the integral over By taking satisfying we guarantee that the sets in (24) are disjoint. Moreover, for any θ 0 ∈ U, by the change of variables and (21) we get Therefore, Further, by (16), a function z → Log κ(z), where Log denotes the principal value of the complex logarithm, is an analytic function in a strip R d + i B where Since for any u ∈ R d we have by Lemma 2.4, there is C > 0 such that for all σ ∈ N d and a + ib ∈ R d + i B, If then for |θ | 2 < we can define Hence, and to finish the proof of theorem it is enough to show Claim 4 Using the integral form for the reminder, we get In view of (26), there is c > 0 such that for all s ∈ R d and θ ∈ B, Therefore, by choosing if |θ | 2 < then we may estimate Next, we write and we split (28) into four corresponding integrals.
Since for a ∈ C, by (29) and (31), the first integrand can be estimated as follows we obtain Furthermore, by (26), which together with (32) implies The third integral is equal zero because the integrand is an odd function. The last one, by (32), we can estimate By putting estimates (33), (34) and (35) together, we obtain Finally, by (8) and Theorem 2.2, there is C > 0 such that for all δ ∈ M and any Hence, which concludes the proof of Claim 4. Although, the asymptotic in Theorem 3.1 is uniform on a large region with respect to n and x, it depends on the implicit function s(δ). By (36), we may estimate and det B s In most applications the following form of the asymptotic of p(n; x) is sufficient.

Remark 1
It is not possible to replace φ(δ) by 1 2 B −1 0 (δ−δ 0 , δ−δ 0 ) without introducing an error term of a very different nature. Indeed, by (12), Since δ 0 ∈ M, if δ approaches ∂M then n|δ − δ 0 | 3 cannot be small. Notice that the third power may be replaced by a higher degree if the random walk has vanishing moments. In particular, for the simple random walk on Z d (see Example 1), for all > 0, x ∈ Z d and n ∈ N, if |x| 1 + n ∈ 2N then otherwise p(n; x) = 0, uniformly with respect to n and x provided that |x| 1 ≤ (1− )n.

Remark 2
It is relatively easy to obtain a global upper bound: for all n ∈ N and x ∈ Z d , p(n; x) ≤ e −nφ(δ) .
Indeed, by Claim 2, for u ∈ R d , we have Hence, by Theorem 2.1, Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Appendix A. Applications
In this section we apply Corollary 3.2 to the simple random walks on triangular and hexagonal lattices.

A.2 The hexagonal lattice
The hexagonal lattice H one may obtain from the triangular lattice by removing all vertices x ∈ L such that τ (x) = 1. Each vertex x ∈ H has three neighbors, Let p be the transition function of the simple random walk on H , i.e. p(x, y) = 1/3 if x and y are closest neighbors. Observe that p is irreducible and periodic with period r = 2. We have where the sum is taken over u ∈ H being a common neighbor of x and y. It is easy to check that q(x, x) = 1/3 and q(x, y) = 1/9 where y belongs to the set If x ∈ X 0 then p(n; 0, x) = q(n/2; 0, x) if n ≡ 0 (mod 2), 0 otherwise.