显式近完全X-Ramanujan图
Explicit near-fully X-Ramanujan graphs
摘要：
设$p(Y_1, \dots, Y_d, Z_1, \dots, Z_e)$是一个自伴非交换多项式，系数来自$\mathbb{C}^{r \times r}$，在待定$Y_1, \dots, Y_d$（被认为是自伴的），待定$Z_1, \dots, Z_e$，以及它们的伴随$Z_1^*, \dots, Z_e^*$中。假设$Y_1, \dots, Y_d$由独立随机$n \times n$匹配矩阵代替，并且$Z_1, \dots, Z_e$由独立随机$n \times n$置换矩阵代替。为了简单起见，假设$p$的系数是$0$-$1$矩阵，则结果可以被认为是一种随机$rn$ -顶点图$G$ .作为$n \to \infty$，将有一个自然的限制无限图$X$，它覆盖了$G$的任何有限结果。Bordenave和Collins最近的一个里程碑式的结果表明，对于任何$\varepsilon > 0$，随机$G$的谱很有可能是$\varepsilon$ -在Hausdorff距离上接近于$X$的谱（一旦排除了适当定义的“平凡”本征值）。我们说$G$是“$\varepsilon$ -接近完全$X$ -Ramanujan”。我们的工作有两个贡献：第一，我们研究和澄清类的无限图形$X$，可以出现在这种方式。第二，我们去随机化的Bordenave柯林斯的结果：对于任何$X$，我们提供明确的，任意大的图$G$所涵盖的$X$和有（非平凡）的频谱在Hausdorff距离最多$\varepsilon$从$X$。这显著地概括了Mohanty等人最近的工作，它提供了明确的近Ramanujan图的每度$d$（意思是$d$ -正则图与所有非平凡的特征值界的幅度由$2\sqrt{d-1} + \varepsilon$）。作为我们的主要技术定理的应用，我们也能够确定的“特征值松弛值”的平均情况下的程度- $2$约束满足问题的广泛的类。
英文摘要：
Let $p(Y_1, \dots, Y_d, Z_1, \dots, Z_e)$ be a self-adjoint noncommutative polynomial, with coefficients from $\mathbb{C}^{r \times r}$, in the indeterminates $Y_1, \dots, Y_d$ (considered to be self-adjoint), the indeterminates $Z_1, \dots, Z_e$, and their adjoints $Z_1^*, \dots, Z_e^*$. Suppose $Y_1, \dots, Y_d$ are replaced by independent random $n \times n$ matching matrices, and $Z_1, \dots, Z_e$ are replaced by independent random $n \times n$ permutation matrices. Assuming for simplicity that $p$'s coefficients are $0$-$1$ matrices, the result can be thought of as a kind of random $rn$-vertex graph $G$. As $n \to \infty$, there will be a natural limiting infinite graph $X$ that covers any finite outcome for $G$. A recent landmark result of Bordenave and Collins shows that for any $\varepsilon > 0$, with high probability the spectrum of a random $G$ will be $\varepsilon$-close in Hausdorff distance to the spectrum of $X$ (once the suitably defined "trivial" eigenvalues are excluded). We say that $G$ is "$\varepsilon$-near fully $X$-Ramanujan". Our work has two contributions: First we study and clarify the class of infinite graphs $X$ that can arise in this way. Second, we derandomize the Bordenave-Collins result: for any $X$, we provide explicit, arbitrarily large graphs $G$ that are covered by $X$ and that have (nontrivial) spectrum at Hausdorff distance at most $\varepsilon$ from that of $X$. This significantly generalizes the recent work of Mohanty et al., which provided explicit near-Ramanujan graphs for every degree $d$ (meaning $d$-regular graphs with all nontrivial eigenvalues bounded in magnitude by $2\sqrt{d-1} + \varepsilon$). As an application of our main technical theorem, we are also able to determine the "eigenvalue relaxation value" for a wide class of average-case degree-$2$ constraint satisfaction problems.
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