Long time behaviour of random walks on the integer lattice

We consider an irreducible finite range random walk on the $d$-dimensional integer lattice and study asymptotic behaviour of its transition function $p(n; x)$. In particular, for simple random walk our asymptotic formula is valid as long as $n (n - |x|_1)^{-2}$ tends to zero.


Introduction
In 1921, with the article [11] Pólya pioneered research on the simple random walk on the integer lattice. Using Fourier analysis he proved that p(n; x), the n'th step transition function, satisfies 1 for any x ∈ Z d . Essentially, Pólya's proof shows that (see Spitzer [ if |x| 1 ≡ n (mod 2), 0 otherwise, uniformly with respect to n ∈ N and x ∈ Z d . As it may be easily seen the local limit theorem is very inaccurate if |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 [ , if |x| 1 ≡ n (mod 2), 0 otherwise, 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 big 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 [ We want to emphasize that the above asymptotic formula is 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 subject of the present article is to give a positive answer to the posed question. Although, the asymptotic will be formulated for the simple random walk, the actual result is valid for any irreducible finite range random walk with the mean zero or not (see Theorem 3.1). Before we state the main theorem, let us introduce some notation. For δ ∈ M = x ∈ R d : |x| 1 < 1 we set where κ is a function on R d defined by In Section 3 we prove the following theorem.
Theorem A. For all x ∈ Z d and n ∈ N, if |x| 1 ≡ n (mod 2) 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(1−|δ| 1 ) 2 stays bounded. Although, the function φ is positive convex and comparable to | · | 2 2 , 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 [2] 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 [2]. Finally, however the quadratic form B x is given explicitly by (7), the mapping M ∋ δ → s(δ) is an implicit function. We want to stress the fact that when |δ| 1 approaches one, 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. Namely, for all ǫ > 0, x ∈ Z d and n ∈ N, if |x| 1 ≤ n(1 − ǫ) then The last asymptotic formula is useful as long as |x| 1 = o(n).
Let us comment about the method of the proof. First, with a 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 analyse by the Laplace method. This is not a straightforward application of it, since the phase function degenerates as |δ| 1 approaches one. To estimate the second part we develop a geometric argument, which allows us to control the way how the quadratic form B s degenerates.
The result obtained in this article has already found an application in the study of subordinated random walks (see [1]) which are spread over all Z d and do not have second moment. Also the geometric method developed here can be successfully applied in much wider context. Namely, to study finitely supported isotropic random walks on affine buildings (see [14]). 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 [3]). Finally, Appendix A contains application of Theorem 3.1 to triangular and hexagonal lattices. This has to be compared with results recently obtained in [6,5,4].
1.1. 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}.

Preliminaries
2.1. 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 and p(1; x) = p(x). The support of p is denoted by V, i.e.
Let κ : C d → C be an exponential polynomial defined by where · , · is the standard scalar product on C d 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 behaviour of transition functions of irreducible 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 d ∈ N we denote the period of p, that is d = gcd n ∈ N : p(n; 0) > 0 .
Then the space Z d decomposes into r disjoint classes X j = x ∈ Z d : p(j + kr; x) > 0 for some k ≥ 0 for j = 0, . . . , r − 1. We observe that for j ∈ {0, . . . , r − 1} and x ∈ X j , if n ≡ j (mod r) then p(n; x) = 0. For each x ∈ Z d , by m x we denote the smallest m ∈ N such that p(m; x) > 0, thus 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 ≥ Kr + m x 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 we may write 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.

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 We conclude that if s is the unique maximum of Let us observe that Since M is not empty, the set V cannot be contained in an affine hyperplane, thus, which implies that lim 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 , δ, u = D u log κ(s).
i.e, D u s = B −1 s u. Therefore, we can calculate . 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 implicit function theorem, the function s is real analytic on M. In particular, s is bounded on any compact subset of M. From the other side, |s| 2 approaches infinity when δ tends to ∂M. To see this, 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 The next theorem provides a control over the speed of convergence in (13). Proof. We consider any enumeration of elements of V = {v 1 , . . . , v N }. Define where S is the unit sphere in R d centred at the origin. Suppose Ω = ∅ and 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 is 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 Ω is a compact set, there is ǫ > 0 such that for all ω ∈ Ω we have Let ∈ Ω and Since If j > k, we can estimate what finishes the proof since by (14) 1 2.3. Analytic lemmas. For a multi-index σ ∈ N d we denote by X σ a multi-set containing σ(i) copies of i. Let Π 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 ∈ Π σ , such that for sufficiently smooth functions f : For a multi-set B containing σ(i) copies of i we set B! = σ!. Let us observe that for the function F (f (s)) is real-analytic in some neighbourhood of s = 0. Hence, there is C > 0 such that for every σ ∈ N d

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 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 behaviour of the n'th step transition density of an irreducible random walk on the integer lattice Z d . Before we state and proof the main theorem, let us present the following example.
Example 3. 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 . Theorem 3.1. Let p be an irreducible 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 Hence, by (5), e int = e i θ0,x = e i(n+r)t , 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 for y ∈ Z d we have Therefore, 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 ǫ > 0 be small enough to satisfy (25), (27) and (30). We set Then the integral over D ǫ d is negligible. To see this, we write Now, we need the following estimate.
For the proof, we assume to contrary that for some v 0 ∈ V and all m ∈ N there In order to apply Claim 3, we select any v 0 satisfying v 0 , s = max v, s : v ∈ V , thus e s,v0 ≥ κ(s). By Claim 3 and (22) Although v may depend on θ, by Theorem 2.2, there are C > 0 and η ≥ 1 such that for all θ ∈ D ǫ Hence, Since n dist(δ, ∂M) η = n 1 2 n dist(δ, ∂M) 2η 1 2 we obtain that e −Cn dist(δ,∂M) η ≤ C ′ n − d 2 −1 dist(δ, ∂M) −2η , provided n is large enough. Finally, since det B s ≤ 1 we conclude that Next, let us consider the integral over By taking ǫ 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 + iB 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 + iB then for |θ| 2 < ǫ we can define ψ(s, θ) = Log κ(s + iθ) − log κ(s) − i θ, δ + 1 2 B s (θ, θ).
Therefore, by choosing if |θ| 2 < ǫ then we may estimate Next, we write and split (28) into four corresponding integrals.
Since for a ∈ C by (29) Furthermore, by (26), which together with (32) implies The third integral is equal zero. 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 In the 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 an higher degree if the random walk has vanishing moments. In particular, for the simple random walk on Z d (see Example 1), for all ǫ > 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 − ǫ)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, p(n; x) ≤ min κ(u) n e − u,x : u ∈ R d ≤ e −nφ(δ) .

Appendix A. Applications
In this section we apply Corollary 3.2 to 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 neighbours, if τ (x) = 2.
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 neighbours. Observe that p is irreducible and periodic with period r = 2. We have where the sum is taken over u ∈ H being a common neighbour 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 x + λ 1 + λ 2 , x + 2λ 1 − λ 2 , x + λ 1 − 2λ 2 , x − λ 1 − λ 2 , x − λ 1 + 2λ 2 , x − 2λ 1 + λ 2 .
Therefore, under the mapping Although, for x ∈ X 1 , we need to apply (39) for three times with different x, the exponential factors are comparable. Indeed, for x ∈ H, x = jλ 1 + j ′ λ 2 let δ be defined by the formula (40). Fix ǫ > 0 and let us consider x ∈ X 1 and n ∈ N such that dist(δ, ∂M) ≥ ǫ.