论文标题
随机平衡的凯利综合体
Random Balanced Cayley Complexes
论文作者
论文摘要
令$ g $为有限的订单$ n $，$ 1 \ leq i \ leq k+1 $ let $ v_i = \ {i \} \ times g $。将每个$ v_i $查看为$ 0 $ -Dimensional复合体，令$ y_ {g，k} $表示simpericial join $ v_1*\ cdots*v_ v_ v_ {k+1} $。对于$ a \ subset g $ ling $ y_ {a，k} $是$ y_ {g，k} $的子复合，其中包含$ y_ {g，k} $的$（k-1）$ - 骨架，$ y_ {g $ k $ -simplices均为$ \ simplices $ \ \ {（1，x_1），x_1），\ ldots，k+1， y_ {g，k} $，以至于$ x_1 \ cdots x_ {k+1} \ in $。令$ l_ {k-1} $表示$ y_ {a，k} $的简化$（k-1）$ - th laplacian of $ y_ {a，k} $，作用于space $ c^{k-1}（y_ _ {a，k}）$的真实价值$（k-1）$的$（k-1）$ - $ y_ {a a，k k} $的cochains $（k-1）$。 $（k-1）$ - th频谱差距$μ_{k-1}（y_ {a，k}）$ y_ {a，k} $是$ l_ {k-1} $的最小特征。证明了以下Alon-Roichman定理的以下$ k $维相似：让$ k \ geq 1 $和$ε> 0 $固定，让$ a $是$ g $的随机子集$ m = $ m = \ weft \ left \ lef lceil \ lceil \ frac {10 k^2 \ log d p} $ rce $ rce $ $ g $的复杂不可还原表示。然后\ [{\ rm pr} \ big [〜μ_ {k-1}（y_ {a，k}）<（1-ε）m〜 \ big] = o \ left（\ frac {1} {n} {n} {n} \ right）。 \]
Let $G$ be a finite group of order $n$ and for $1 \leq i \leq k+1$ let $V_i=\{i\} \times G$. Viewing each $V_i$ as a $0$-dimensional complex, let $Y_{G,k}$ denote the simplicial join $V_1*\cdots*V_{k+1}$. For $A \subset G$ let $Y_{A,k}$ be the subcomplex of $Y_{G,k}$ that contains the $(k-1)$-skeleton of $Y_{G,k}$ and whose $k$-simplices are all $\{(1,x_1),\ldots,(k+1,x_{k+1})\} \in Y_{G,k}$ such that $x_1\cdots x_{k+1} \in A$. Let $L_{k-1}$ denote the reduced $(k-1)$-th Laplacian of $Y_{A,k}$, acting on the space $C^{k-1}(Y_{A,k})$ of real valued $(k-1)$-cochains of $Y_{A,k}$. The $(k-1)$-th spectral gap $μ_{k-1}(Y_{A,k})$ of $Y_{A,k}$ is the minimal eigenvalue of $L_{k-1}$. The following $k$-dimensional analogue of the Alon-Roichman theorem is proved: Let $k \geq 1$ and $ε>0$ be fixed and let $A$ be a random subset of $G$ of size $m= \left\lceil\frac{10 k^2\log D}{ε^2}\right\rceil$ where $D$ is the sum of the degrees of the complex irreducible representations of $G$. Then \[ {\rm Pr}\big[~μ_{k-1}(Y_{A,k}) < (1-ε)m~\big] =O\left(\frac{1}{n}\right). \]
