正则子图密度的上尾变分问题中的双块优化子
Bipodal optimizers in the upper-tail variational problem for regular subgraph densities
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中文总结 AI 辅助
研究正则子图密度的上尾变分问题，证明在对称破缺侧优化子为双块且唯一，并分析其收敛机制与渐近性质。
中文摘要 AI 辅助
设 $H$ 为固定的 $d$-正则图，其中 $d\ge2$，并令 $t(H,\cdot)$ 表示其同态密度。我们研究稠密 Erdős--Rényi 随机图 $G(n,p)$ 中固定 $0<p<r<1$ 的上尾事件 $t(H,G(n,p))\ge r^{|E(H)|}$。在 Lubetzky--Zhao 复制对称相边界附近且远离例外目标密度 $(d-1)/d$ 时，我们证明 Chatterjee--Varadhan 变分问题在对称破缺侧的优化子是双块（两块）的，并且在重新标记意义下唯一。其块参数关于 $(p,r)$ 解析。为处理例外边界点（非例外理论在此退化），我们构造一条从对称破缺侧逼近该点的解析曲线，沿此曲线唯一优化子是非常数秩一双块图元。在两种情形下，我们推导了边密度亏缺和速率函数的渐近展开，该速率函数控制上尾概率的指数衰减。此外，当 $n\to\infty$ 时，条件随机图在割距离下收敛到相应的双块优化子。当 $(p,r)$ 从对称破缺侧接近相边界时，优化子通过两种不同机制收敛到其常数极限。对于每个固定的非例外目标密度，一个块收缩到零测度，给出 $L^1$ 收敛但不给出 $L^\infty$ 收敛。沿例外曲线，两个块保持宏观：其大小趋于 $1/2$，且所有三个块密度趋于 $(d-1)/d$，给出 $L^\infty$ 收敛。
英文摘要
Let $H$ be a fixed $d$-regular graph with $d\ge2$, and let $t(H,\cdot)$ denote its homomorphism density. We study the upper-tail event $t(H,G(n,p))\ge r^{|E(H)|}$ for fixed $0<p<r<1$ in a dense Erdős--Rényi random graph $G(n,p)$. Near the Lubetzky--Zhao replica-symmetric phase boundary and away from the exceptional target density $(d-1)/d$, we prove that the optimizer of the Chatterjee--Varadhan variational problem on the symmetry-breaking side is bipodal (two-block) and unique up to relabeling. Its block parameters depend analytically on $(p,r)$. To treat the exceptional boundary point, where the nonexceptional theory degenerates, we construct an analytic curve approaching that point from the symmetry-breaking side along which the unique optimizers are nonconstant rank-one bipodal graphons. In both settings, we derive asymptotic expansions of the edge-density deficit and the rate function that governs the exponential decay of the upper-tail probability. Moreover, the conditioned random graph converges in cut distance to the corresponding bipodal optimizer as $n\to\infty$. As $(p,r)$ approaches the phase boundary from the symmetry-breaking side, the optimizers converge to their constant limits through two distinct mechanisms. For each fixed nonexceptional target density, one block shrinks to zero measure, giving convergence in $L^1$ but not in $L^\infty$. Along the exceptional curve, both blocks remain macroscopic: their sizes tend to $1/2$ and all three block densities tend to $(d-1)/d$, yielding convergence in $L^\infty$.
发表机构
- School of Computing, KAIST（KAIST计算机学院）
- Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院）
- Center for AI and Natural Sciences, Korea Institute for Advanced Study (KIAS)（韩国高等科学研究院人工智能与自然科学中心）
- Department of Mathematical Sciences, KAIST（KAIST数学科学系）
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