The authors study quantitatively the effective large-scale behaviour of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. If the coefficients \(a(\cdot)\) are stationary and ergodic, then a qualitative homogenization result is known which says that as \(\varepsilon \downarrow0\) the elliptic operator \(-\nabla\cdot a(\cdot/\varepsilon)\nabla\) almost surely \(H\)-converges to the homogenized elliptic operator \(-\nabla\cdot a_{\text{hom}}\nabla\). The authors consider linear second-order difference equations with uniformly elliptic, bounded, diagonal random coefficients. Denote by \(\Omega_0=\{\text{diag}(a_1,\ldots,a_{d})\in \mathbb R^{d\times d}:\;\lambda\leq a_{j}\leq 1, j=1,\dots,d\}\) the set of admissible coefficient matrices, where \(\lambda>0\) is an ellipticity constant. A coefficient field, denoted by \(a\), is a function on \(\mathbb Z^{d}\) taking values in \(\Omega_0\). The authors endow \(\Omega=(\Omega_0)^{\mathbb Z^{d}}\) with the product topology. A probability measure on \(\Omega\) is called an ensemble and denote the associated ensemble average by \(\langle\cdot\rangle\).
For scalar fields \(\zeta:\;\mathbb Z^{d}\to \mathbb R\), vector fields \(\xi=(\xi_1,\dots,\xi_{d}):\;\mathbb Z^{d}\to \mathbb R^{d}\), the spatial derivatives \(\nabla_{i}\zeta(x)=\zeta(x+e_ i)-\zeta(x)\), \(\nabla\zeta=(\nabla_1\zeta,\dots,\nabla_{d}\zeta)\), \(\nabla_{i}^{*}\zeta(x)=\zeta(x-e_ i)-\zeta(x)\), \(\nabla^{*}\xi=\sum_{i=1}^{d}\nabla_{i}^{*}\xi_{i}\) are defined, where \((e_1,\dots,e_{d})\) denotes the canonical basis of \(\mathbb R^{d}\).
For scalar random variables \(\zeta:\;\Omega\to \mathbb R\), vector-valued random variables \(\xi=(\xi_1,\dots,\xi_{d}):\;\Omega\to \mathbb R^{d}\) let us define the horizontal derivatives \(D_{i}\zeta(a)=\zeta(a(\cdot+e_ i))-\zeta(a)\), \(D\zeta=(D_1\zeta,\dots,D_{d}\zeta)\), \(D^{*}_{i}\zeta(a)=\zeta(a(\cdot-e_{i})-\zeta(a)\), \(D^{*}\xi=\sum_{i=1}^{d}D^{*}_{i}\xi_{i}\).
With \(\langle\cdot\rangle\) the symmetric matrix of the homogenized coefficients \(a_{hom}\) via the minimization problem \(\forall e\in \mathbb R^{d}:\;e\cdot a_{hom}e=\inf_{\bar\phi}\langle(e+\nabla\bar\phi)\cdot a(e+\nabla\bar\phi)\rangle\) is associated, where the infimum runs over all stationary random fields \(\bar\phi:\;\Omega\times \mathbb Z^{d}\to \mathbb R\). The corrector equation has the form \(\nabla^{*}a(\nabla\bar\phi+e)=0\) on \(\mathbb Z^{d}\) and the homogenization formula \(a_{hom}e=\langle a(\nabla\bar\phi+e)\rangle\) holds true if a stationary corrector exists.
We say that \(\langle\cdot\rangle\) satisfies a spectral gap with constant \(\rho>0\) on the torus of size \(L\in \mathbb N\), in short \(SG_{L}(\rho)\), if for all \(\zeta\in L^2(\Omega)\) with \(\langle\zeta\rangle =0\) we have \(\langle\zeta^2\rangle\leq {1\over\rho}\sum_{y\in([0,L]\cap \mathbb Z)^{d}}\left\langle(\partial\zeta/\partial_{L}y)^2\right\rangle\), where \(\partial\zeta/\partial_{L}y=\zeta-\langle\zeta\rangle_{L,y}\), \(\langle\zeta\rangle_{L,y}\) is the conditional expectation of \(\zeta\) given \(a(x)\) for all \(x\in \mathbb Z^{d}\setminus\{y+L\mathbb Z^{d}\}\).
The main result of this paper is the following. Assume that \(\langle\cdot\rangle\) is stationary, \(L\)-periodic and satisfies \(SG_{L}(\rho)\). Then there exists an exponent \(1\leq p_0<\infty\) that only depends on the \(\lambda\) and \(d\geq2\) such that the following statement holds for every \(p\in(p_0,\infty)\):
For \(\xi\in C_{b}(\Omega)^{d}\) and \(t\geq0\) let us define \(u(t)=\exp(-tD^{*}a(0)D)D^{*}\xi\). Then we have \[ \langle| u(t)|^{2p}\rangle^{1/2p}\leq C(t+1)^{-({d\over4}+{1\over2})}\|\partial\xi\|_{l_{y}^1 L_{\langle\cdot\rangle}^{2p}}, \text{ where } \|\partial\xi\|_{l_{y}^1 L_{\langle\cdot\rangle}^{2p}}=\sum_{y\in([0,L]\cap \mathbb Z)^{d}}\left\langle\left|{\partial\xi\over\partial_{L}y}\right|^{2p}\right\rangle^{1/2p}, \] constant \(C\) depends only on \(p,\rho,\lambda\) and \(d\).