Among the models of disordered conduction and localization, models with $N$ orbitals per site are attractive both for their mathematical tractability and for their physical realization in coupled disordered grains. However Wegner proved that there is no Anderson transition and no localized phase in the $N\to \infty$ limit, if the hopping constant $K$ is kept fixed (Wegner, 1979; Khorunzhy and Pastur, 1993). Here we show that the localized phase is preserved in a different limit where $N$ is taken to infinity and the hopping $K$ is simultaneously adjusted to keep $NK$ constant. We support this conclusion with two arguments. The first is numerical computations of the localization length showing that in the $N\to \infty$ limit the site-diagonal-disorder model possesses a localized phase if $NK$ is kept constant, but does not possess that phase if $K$ is fixed. The second argument is a detailed analysis of the energy and length scales in a functional integral representation of the gauge invariant model. The analysis shows that in the $K$ fixed limit the functional integral’s spins do not exhibit long distance fluctuations, i.e. such fluctuations are massive and therefore decay exponentially, which signals conduction. In contrast the $NK$ fixed limit preserves the massless character of certain spin fluctuations, allowing them to fluctuate over long distance scales and cause Anderson localization.

在无序传导和局部化的模型中，具有 $ñ$每个位点的轨道因其数学上的易处理性以及它们在耦合无序晶粒中的物理实现而具有吸引力。然而，韦格纳证明，在该过程中没有Anderson过渡，也没有局部相。$ñ\to \infty$ 极限，如果跳变常数 $ķ$保持固定（Wegner，1979； Khorunzhy和Pastur，1993）。在这里，我们显示了本地化阶段保留在不同的限制中，其中$ñ$ 被带到无限和跳跃 $ķ$ 同时调整以保持 $ñķ$不变。我们用两个论据支持这一结论。首先是定位长度的数值计算，表明在$ñ\to \infty$ 限制位点-对角线紊乱模型具有局部相位，如果 $ñķ$ 保持恒定，但不具有该相位，如果 $ķ$是固定的。第二个参数是量规不变模型的功能积分表示中的能量和长度尺度的详细分析。分析表明，在$ķ$固定极限时，功能积分的自旋不会表现出长距离波动，即这种波动很大，因此呈指数衰减，这表示传导。相反，$ñķ$ 固定极限保留了某些自旋波动的无质量特征，使它们可以在远距离范围内波动并引起安德森局部化。