We slightly modify the proof of Hanson-Wright inequality (HWI) for concentration of Gaussian quadratic chaos where we tighten the bound by increasing the absolute constant in its formulation from the largest known value of 0.125 to at least 0.145 in the symmetric case. We also present a sharper version of an inequality due to Laurent and Massart (LMI) through which we increase the absolute constant in HWI from the largest available value of approximately $0.134$ due to LMI itself to at least $0.152$ in the positive-semidefinite case. A new sequence of concentration bounds indexed by $m=1,2,3,\cdots, \infty$ is developed that involves Schatten norms of the underlying matrix. The case $m=1$ recovers HWI. These bounds undergo a phase transition in the sense that if the tail parameter is smaller than a critical threshold $τ_c$, then $m=1$ is the tightest and if it is larger than $τ_c$, then $m=\infty$ is the tightest. This leads to a novel bound called the~$m_\infty$-bound. A separate concentration bound named twin to HWI is also developed that is tighter than HWI for both sufficiently small and large tail parameter. Finally, we explore concentration bounds when the underlying matrix is positive-semidefinite and only the dimension~$n$ and its largest eigenvalue are known. Five candidates are examined, namely, the $m_\infty$-bound, relaxed versions of HWI and LMI, the $χ^2$-bound and the large deviations bound. The sharpest among these is always either the $m_\infty$-bound or the $χ^2$-bound. The case of even dimension is given special attention. If $n=2,4,6$, the $χ^2$-bound is tighter than the $m_\infty$-bound. If $n$ is an even integer greater than or equal to 8, the $m_\infty$-bound is sharper than the $χ^2$-bound if and only if the ratio of the tail parameter over the largest eigenvalue lies inside a finite open interval which expands indefinitely as $n$ grows.

翻译：我们略微修改了高斯二次混沌浓度的 Hanson-Wright 不等式（HWI）证明，通过将其表述中的绝对常数从已知最大值 0.125 提高至至少 0.145（对称情形），从而收紧界。我们还通过改进 Laurent 和 Massart 的不等式（LMI），在正半定情形下将 HWI 的绝对常数从 LMI 本身提供的约 $0.134$ 提升至至少 $0.152$。我们发展了一个由 $m=1,2,3,\\cdots, \\infty$ 索引的浓度界序列，其中涉及底层矩阵的 Schatten 范数。当 $m=1$ 时，该序列恢复 HWI。这些界经历相变：若尾部参数小于临界阈值 $\\tau_c$，则 $m=1$ 的界最紧；若大于 $\\tau_c$，则 $m=\\infty$ 的界最紧。这引出了一个称为 $m_\\infty$-界的新界。我们还提出了一个与 HWI 成对的独立浓度界，该界在尾部参数足够小和足够大时均比 HWI 更紧。最后，我们探讨了当底层矩阵为正半定且仅已知维度 $n$ 及其最大特征值时的浓度界。我们检验了五个候选界：$m_\\infty$-界、HWI 和 LMI 的松弛版本、$\\chi^2$-界以及大偏差界。其中最锐利的总是 $m_\\infty$-界或 $\\chi^2$-界。我们对偶数维情形给予了特别关注。若 $n=2,4,6$，则 $\\chi^2$-界比 $m_\\infty$-界更紧。若 $n$ 为大于等于 8 的偶数，则当且仅当尾部参数与最大特征值的比值位于一个有限开区间内时，$m_\\infty$-界比 $\\chi^2$-界更锐利，且该区间随 $n$ 增大而无限扩展。