We prove non-perturbative bounds on the time evolution of the probability distribution of operator size in the $q$-local Sachdev-Ye-Kitaev model with $N$ fermions, for any even integer $q>2$ and any positive even integer $N>2q$. If the couplings in the Hamiltonian are independent and identically distributed Rademacher random variables, the infinite temperature many-body Lyapunov exponent is almost surely finite as $N\rightarrow\infty$. In the limit $q \rightarrow \infty$, $N\rightarrow \infty$, $q^{6+\delta}/N \rightarrow 0$, the shape of the size distribution of a growing fermion, obtained by leading order perturbation calculations in $1/N$ and $1/q$, is similar to a distribution that locally saturates our constraints. Our proof is not based on Feynman diagram resummation; instead, we note that the operator size distribution obeys a continuous time quantum walk with bounded transition rates, to which we apply concentration bounds from classical probability theory.

中文翻译：

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Sachdev-Ye-Kitaev 模型中算子大小分布的非微扰动力学

我们证明了 $q$-local Sachdev-Ye-Kitaev 模型中算子大小的概率分布的时间演化的非微扰边界和 $N$ 费米子，对于任何偶数 $q>2$ 和任何正偶数$N>2q$。如果哈密顿量中的耦合是独立同分布的 Rademacher 随机变量，则无限温度多体 Lyapunov 指数几乎肯定是有限的 $N\rightarrow\infty$。在极限 $q \rightarrow \infty$, $N\rightarrow \infty$, $q^{6+\delta}/N \rightarrow 0$ 中，生长费米子的尺寸分布形状，通过先导顺序获得$1/N$ 和 $1/q$ 中的扰动计算类似于局部饱和我们的约束的分布。我们的证明不是基于费曼图的归纳；反而，