On the behavior of random walk around heavy points
The random walk \(S_{0}=0,S_n=\sum_{k=1}^n X_k,n=1,2,\dots \) on the d-dimensional integer lattice \(\mathbb{Z}^d\) is considered where \(X_k\), \(k=1,2,\dots \) are i.i.d. random variables with the distribution \(P\{ X_{1}=x\} =p(x) ,x\in \mathbb{Z}^d\) such that \(\sum_{x\in \mathbb{Z}^d}| x|^{2}p(x) dx<\infty ,d=3\), \(\sum_{x\in \mathbb{Z}^d}| x|^{2}\log (| x| +1) p(x) <\infty ,d=4\), \(\sum_{x\in \mathbb{Z}^d}| x|^{d-2}p(x) dx<\infty ,d\geq5\). The covariance matrix of \(X\) is \(Q\). \(| Q| \) is its determinant and \(Q^{-1}\) is its inverse. The random walk \(\{ S_n\} \) is supposed to be aperiodic. Its Green function \(G( x) :=\sum_{n=0}^\infty P\{Sn=x\} ,x\in \mathbb{Z}^d\) has the asymptotic property \(G( x) \sim C_d| Q|^{-\frac{1}{2} }\| x\|^{2-d}\) as \(| x| \to \infty ,\| x\|^{2}=xQ^{-1}x\), with a certain constant \(C_d\). Denote \(\gamma =\frac{1}{G( 0)};1-\gamma_{x}=\frac{G( x) }{G( 0) },x\neq 0\). The local time is as follows \(\xi ( x,n) :=\sum_{k=1}^n I\{ S_k=x\} ,n=1,2,\dots ,x\in \mathbb{Z}^d\). The random walk is transient for \(d\geq 3\), i.e. any point can have only a finite number of visits, but there are (random) points that accumulate a higher number of visits, the so-called heavy points. The problem is to find out wheather there can be another heavy point or a point not visited at all up to time \(n\) in the close neighborhood of a heavy point. The main statements in this paper imply the negative answer. Denote \(\xi ( n) :=\max_{x\in \mathbb{Z}^d}\xi ( x,n) ,n=1,2,\dots \), \(\eta ( n) :=\max_{1\leq j\leq n}\xi ( S_{j},\infty ) ,n=1,2,\dots\). Even for the general aperiodic random walk \(\lim_{n\to \infty }\frac{\eta ( n) }{\log n} =\lambda \) a.s. with \(\lambda =-\frac{1}{\log ( 1-\gamma ) }\). Define \[ m_{x}=\begin{cases} 1&\text{if }x<0 \\ \frac{( 1-\gamma _{x}) ^{2}}{1-\gamma}& \text{if }x\neq 0.\end{cases} \] Let \(k_n=( 1-\delta _n) \lambda \log n\), where \(\delta _n>0\) is non-increasing. Let \(r_n>0\) be non-decreasing and for any \(c>0\) let \(r_{ [ cn]}/r_n<C\). Let for \(\beta _n:=r_n^{2d-4}\frac{\log \log n}{\log (n)},\lim_{n\to \infty }\beta _n=0\), \(\lim_{n\to \infty}\delta _nr_n^{2d-4}=0\). Define \(A_n=\{ z\in \mathbb{Z}^d:\xi (z,n)\geq k_n\} ,\) \(B(r)=\{ x:\| x\| \leq r\}\), \(B_n=\{ j\leq n:\xi (S_{j},\infty )\geq k_n\}\). For the symmetric aperiodic random walk (Th. 1.1) \(\lim_{n\to \infty }\sup_{z\in A_n} \sup_{x\in B(r_n)}| \frac{\xi ( z+x,n) }{ m_{x}\lambda \log n}-1| =0\) a.s., \(d\geq 5\) and for \(d\geq 3\) (Th. 1.2) \(\lim_{n\to \infty }\sup_{j\in B_n} \sup_{x\in B(r_n)}| \frac{\xi ( S_{j}+x,\infty )}{m_{x}\lambda \log n}-1| =0\) a.s. If \(z\) belongs to a fixed set \(A\), the corresponding corollaries are presented. For \(d\geq 3,j\in B_n\) let \(R(n,j)\) denote the number such that \(\xi (S_{j}+x,\infty )>0,x\in B(R(n,j))\). Corollary 1.2 states that \(R(n,j)\geq (\log n)^{\frac{1-\varepsilon }{2d-4}}\) almost surely. The authors conjecture that for \(j\in B_n,R(n,j)\leq (\log n)^{\frac{1+\varepsilon }{ 2d-4}}\). Th.1.3 proved for simple random walk, \(d\geq 3\), shows that in Th. 1.2 the power \(\frac{1}{2d-4}\) of \(\log n\) cannot be improved in general.
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