Airy limit for $β$-additions through Dunkl operators

Abstract

It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of $β$-ensembles is given by the $\mathrm{Airy}(β)$ point process. We extend this universality result to a general class of additions of Gaussian and Laguerre ensembles, which were identified in \cite{AN} as projection of the ergodic measures of the $β$-corners process. In order to make sense of the $β$-addition, we introduce the Type-A Bessel function as the characteristic function of our matrix ensemble, following the approach of \cite{GM}, \cite{BCG}. Then we extract its moment information through the action of Dunkl operators, and obtain certain limiting functional expressed via conditional Brownian bridges for the Laplace transform of $\mathrm{Airy}(β)$. Our limit expression is universal up to proper rescaling among all of our additions, and agrees with the single-time Laplace transform expression from the concurrent work \cite{GXZ}.

Figures (4)

• Figure 1: The three cases for the paths in Theorem \ref{['thm:brownian2']}.
• Figure 2: An illustration of the classification of main variables into types I-VI. In each figure the walk $E(t)$ is drawn in black, the sum of blocks $q(t)$ is drawn in blue, and a particular block $q_j$ is drawn in red if relevant. Top: The left shows Type I in which $\tau_i <\tilde{\tau}_i$. The right shows Types II and III in which $\tau_i=\tilde{\tau}_i$. The difference between the two types depends on whether the block $q_j$, drawn in red, belongs to an auxiliary variable (Type II) or a main variable (Type III). Middle: The left shows Type IV in which $\tilde{\tau}_i<\tau_i$. The right shows Type V in which both $\tau_i$ and $\tilde{\tau}_i$ are $\varnothing$. Note that the illustrations show that $E(C_{a_i-1}) > q(C_{a_i-1})$ but $E(C_{a_i-1}) = q(C_{a_i-1})$ is also allowed for these types. Bottom: The illustrations for the three subtypes of Type VI. In each of the subtypes $\tau_i=\tilde{\tau}_i$. In Type VI.1, the block $q_j$ drawn in red can be any main or auxiliary variable. In Type VI.3 the red block $q_i$ represents variable $x_i$.
• Figure 3: An illustration of Types A and B. In both cases the block $q_i$ is drawn in red.
• Figure 4: An example of the deformation algorithm given in Section \ref{['sec:secondcancellation']}. In each configuration black is the walk, blue is the sum of the blocks, red is the block for $x_1$, and green is the block for $x_{i_2'}$. The labels above each interval denote the index of the corresponding main variable. Top: We start with a configuration in $\mathcal{B}(1,i_{2}',2,i_{4}')$ where $i_{2}'\ne i_{4}'$ are two distinct elements in $\llbracket 3,N\rrbracket$. Note that $x_{i_2'}$ is of Type III with $x_1$ the corresponding variable of Type B, and $x_{i_4'}$ is of Type III with $x_{i_2'}$ the corresponding variable of Type B. To deform the configuration, we start with the rightmost Type III variable and delete the jump of $x_{i_2'}$ between $C_3$ and $C_4$, then change the main variable for this interval to $x_{i_2'}$. Middle: After the first deformation, $x_{i_2'}$ is still of Type III with $x_1$ the corresponding variable of Type B. We deform the configuration by deleting the jump of $x_1$ between $C_1$ and $C_2$ and changing all instances of the variable $x_{i_2'}$ to $x_1$. This changes the main variable in the intervals from $C_1$ to $C_2$ and $C_3$ to $C_4$ from $x_{i_2'}$ to $x_1$, as well as changing the block of $x_{i_2'}$ to a block of $x_1$. Bottom: The end result is a configuration in $\mathcal{B}(1,1,2,1)$ without any Type III variables.

Theorems & Definitions (73)

• Proposition 1.1
• Theorem 1.2: Edge universality of $\beta$-ensemble additions
• Proposition 1.3
• Remark 1.4
• Remark 1.5
• Example 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4