论文标题
一类带有远距离跳跃的随机电导模型的猝灭不变性原理
Quenched Invariance Principle for a class of random conductance models with long-range jumps
论文作者
论文摘要
我们在固定的ergodic随机电导中研究$ \ mathbb z^d $（带有$ d \ ge 2 $）上的随机步行，$ \ {c_ {c_ {x，y} \ colon x，y \ in \ in \ mathbb z^d \} $允许任意长度跳高。我们的重点是我们通过校正方法，功能不平等和热界技术的结合来确定的猝灭不变性原理（QIP），假设$ p $ -th的矩$ \ sum_ {x \ in \ sum_ {x \ in \ sumbb z^d} $ p，q \ ge1 $的原点是有限的，带有$ p^{ - 1}+q^{ - 1} <2/d $。尤其是，QIP在所有$ d \ ge2 $中的连接指数大于$ 2D $的远程渗透图上随机步行，只要所有最近的邻居边缘都存在。尽管仍然受到瞬间条件的限制，但我们的证明方法是新颖的，因为它避免了校正者的各个公共性证明。这很相关，因为我们表明，对于长期渗透，指数在$ d+2 $和$ 2D $之间，校正器存在，但无处不在。在与P. Bella和M.Schäffner最近工作的条件下，还为最近的邻居，千古电导量构建了类似的示例。这些例子阐明了这些问题最近进展的许多进展的基础椭圆形技术的局限性。
We study random walks on $\mathbb Z^d$ (with $d\ge 2$) among stationary ergodic random conductances $\{C_{x,y}\colon x,y\in\mathbb Z^d\}$ 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 $p,q\ge1$ 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 $d\ge2$, 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 $d\ge3$ under the conditions complementary to those of the recent work of P. Bella and M. Schäffner. These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.
