arXiv 1609.093742016-09-29math.PRmath.CO

单模随机树的谱原子

Spectral atoms of unimodular random trees

Justin Salez

摘要：

我们利用质量传输原理分析了么模随机树邻接算子的预解式$(A-z)^{-1}$的局部递归。在复杂的参数$z$接近一个给定的位置$λ$的实轴上的限制，我们表明，这种递归诱导分解的树成有限块的几何形状直接确定的光谱质量在$λ$。然后，我们利用这种对应关系，以获得精确的信息的纯点支持的频谱，在扩展性能的树。特别地，我们推导出任何具有最小度$δ\ge 3$和最大度$Δ$的幺模随机树的谱的纯点支撑被限制为100个点，即大小小于$\frac{Δ-2}{δ-2}$的树的特征值。更一般地说，我们表明，限制$δ\ge 3$可以被削弱到$δ\ge 2$，只要锚定的等周常数的树保持有界远离$0$。这特别适用于任何没有叶子的幺模高尔顿-沃森树，允许我们解决Bordenave，Sen和Virág（2013）的猜想。

英文摘要：

We use the Mass Transport Principle to analyze the local recursion governing the resolvent $(A-z)^{-1}$ of the adjacency operator of unimodular random trees. In the limit where the complex parameter $z$ approaches a given location $λ$ on the real axis, we show that this recursion induces a decomposition of the tree into finite blocks whose geometry directly determines the spectral mass at $λ$. We then exploit this correspondence to obtain precise information on the pure-point support of the spectrum, in terms of expansion properties of the tree. In particular, we deduce that the pure-point support of the spectrum of any unimodular random tree with minimum degree $δ\ge 3$ and maximum degree $Δ$ is restricted to finitely many points, namely the eigenvalues of trees of size less than $\frac{Δ-2}{δ-2}$. More generally, we show that the restriction $δ\ge 3$ can be weakened to $δ\ge 2$, as long as the anchored isoperimetric constant of the tree remains bounded away from $0$. This applies in particular to any unimodular Galton-Watson tree without leaves, allowing us to settle a conjecture of Bordenave, Sen and Virág (2013).

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