We study random walks on \({\mathbb {Z}}^d\) (with \(d\ge 2\)) among stationary ergodic random conductances \(\{C_{x,y}:x,y\in {\mathbb {Z}}^d\}\) that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the p-th moment of \(\sum _{x\in {\mathbb {Z}}^d}C_{0,x}|x|^2\) and q-th moment of \(1/C_{0,x}\) for x neighboring the origin are finite for some \(p,q\ge 1\) with \(p^{-1}+q^{-1}<2/d\). In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than 2d in all \(d\ge 2\), provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between \(d+2\) and 2d, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in \(d\ge 3\) under the conditions complementary to those of the recent work of Bella and Schäffner (Ann Probab 48(1):296–316, 2020). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.

我们研究了 \({\mathbb {Z}}^d\) (with \(d\ge 2\) ) 在平稳遍历随机电导\(\{C_{x,y}:x,y\in {\mathbb {Z}}^d\}\)允许任意长度的跳转。我们的重点是在淬火不变性原理（QIP），我们通过的修正方法，泛函不等式和热核心技术假设组合建立的p的第时刻\（\总和_ {X \在{\ mathbb {Z }}^d}C_{0,x}|x|^2\)和\(1/C_{0,x}\) 的q阶矩对于 与原点相邻的x是有限的，对于某些\(p,q \ge 1\)与\(p^{-1}+q^{-1}<2/d\). 特别是，QIP 因此适用于在所有\(d\ge 2\) 中具有大于 2 d 的连通性指数的远程渗透图上的随机游走，前提是所有最近邻边都存在。尽管仍然受到矩条件的限制，但我们的证明方法是新颖的，因为它避免了证明校正器的处处次线性。这是相关的，因为我们表明，对于指数在\(d+2\)和 2 d之间的长程渗透，校正器存在但不能在任何地方都是次线性的。也为\(d\ge 3\) 中的最近邻、遍历电导构造了类似的例子在与 Bella 和 Schäffner 最近的工作（Ann Probab 48(1):296–316, 2020）互补的条件下。这些示例阐明了椭圆正则性技术的局限性，而这些局限性正是这些问题最近取得的大部分进展的基础。