连接Eigenvalue刚性与聚合物几何：大偏差下的扩散横向涨落
Connecting Eigenvalue Rigidity with Polymer Geometry: Diffusive Transversal Fluctuations under Large Deviation
摘要：
在$\mathbb{Z}^2$上考虑了大偏差区域下指数定向末次通过渗流模型的精确可解性。在上尾大偏差事件$\mathcal{U}_δ:=\{T_{n}\geq (4+δ)n\}$的条件下，其中$T_{n}$表示从$(1,1)$到$(n,n)$的最后通过时间，我们研究了聚合物/测地线$Γ_{n}$的几何形状，即，最优路径达到$T_{n}$。我们表明，条件作用于$\mathcal{U}_δ$会将横向涨落指数从KPZ普适类的特征$2/3$改变为$1/2$，即，有条件地，包含$Γ_{n}$的对角线周围的最小条带具有高概率的宽度$n^{1/2+o(1)}$。这使Deuschel和Zeitouni（1999）的结果更加尖锐，他们证明了在泊松最后一次通过渗流的背景下横向波动的$o(n)$界限，并补充了（Basu，Ganguly，Sly，2017），其中横向波动被证明是下尾大偏差事件中的$Θ(n)$。我们的证明利用指数LPP模型中的最后通过时间与拉盖尔酉包络（LUE）的最大特征值之间的对应关系以及后者的谱的行列式结构。在我们的证明中的一个关键因素是一个尖锐的最大特征值（Seppäläinen '98，Johansson '99）的大偏差结果的改进，使用刚性的频谱，这可能是独立的利益。
英文摘要：
We consider the exactly solvable model of exponential directed last passage percolation on $\mathbb{Z}^2$ in the large deviation regime. Conditional on the upper tail large deviation event $\mathcal{U}_δ:=\{T_{n}\geq (4+δ)n\}$ where $T_{n}$ denotes the last passage time from $(1,1)$ to $(n,n)$, we study the geometry of the polymer/geodesic $Γ_{n}$, i.e., the optimal path attaining $T_{n}$. We show that conditioning on $\mathcal{U}_δ$ changes the transversal fluctuation exponent from the characteristic $2/3$ of the KPZ universality class to $1/2$, i.e., conditionally, the smallest strip around the diagonal that contains $Γ_{n}$ has width $n^{1/2+o(1)}$ with high probability. This sharpens a result of Deuschel and Zeitouni (1999) who proved a $o(n)$ bound on the transversal fluctuation in the context of Poissonian last passage percolation, and complements (Basu, Ganguly, Sly, 2017), where the transversal fluctuation was shown to be $Θ(n)$ in the lower tail large deviation event. Our proof exploits the correspondence between last passage times in the exponential LPP model and the largest eigenvalue of the Laguerre Unitary Ensemble (LUE) together with the determinantal structure of the spectrum of the latter. A key ingredient in our proof is a sharp refinement of the large deviation result for the largest eigenvalue (Seppäläinen '98, Johansson '99), using rigidity properties of the spectrum, which could be of independent interest.
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