论文标题
在稀疏的随机组合矩阵上
On sparse random combinatorial matrices
论文作者
论文摘要
令$ q_ {n，d} $表示行的随机组合矩阵的行彼此独立，因此每行都是从$ \ {0,1 \}^n $中的向量子集中均匀地采样的，其精确$ d $ d $ cotties cock {0,1 \}^n $。我们提供了一个简短的证明，即$ \ pr [\ det（q_ {n，d}）= 0] = 0] = o \ left（\ frac {n^{1/2} \ log^{3/2} n} n} n} n} n} {d} {d} {d} {d} {d} {d} {d} {d} = o（1）特别是，我们的证明可以容纳稀疏的随机组合矩阵，从某种意义上说，允许$ d = o（n）$。 我们还考虑了确定性整数矩阵的奇异性$ a $由稀疏组合矩阵随机扰动。特别是，我们证明了$ \ pr [\ det（a+q_ {n，d}）= 0] = 0] = o \ left（\ frac {n^{1/2} \ log^{3/2} n} n} n} n} {d} {d} {d} {d} \ right） $（1，-d）$不是$ a $的特征。
Let $Q_{n,d}$ denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in $\{0,1\}^n$ having precisely $d$ entries equal to $1$. We present a short proof of the fact that $\Pr[\det(Q_{n,d})=0] = O\left(\frac{n^{1/2}\log^{3/2} n}{d}\right)=o(1)$, whenever $d=ω(n^{1/2}\log^{3/2} n)$. In particular, our proof accommodates sparse random combinatorial matrices in the sense that $d = o(n)$ is allowed. We also consider the singularity of deterministic integer matrices $A$ randomly perturbed by a sparse combinatorial matrix. In particular, we prove that $\Pr[\det(A+Q_{n,d})=0]=O\left(\frac{n^{1/2}\log^{3/2} n}{d}\right)$, again, whenever $d=ω(n^{1/2}\log^{3/2} n)$ and $A$ has the property that $(1,-d)$ is not an eigenpair of $A$.
