Independent GUE minor processes of perfect matchings on rail-yard graphs

Abstract

We study perfect matchings on the rail-yard graphs in which the right boundary condition is given by the empty partition and the left boundary can be divided into finitely many alternating line segments where all the vertices along each line segment are either removed or remained. When the edge weights satisfy certain conditions, we show that the distributions of the locations of certain types of dimers near the right boundary converge to the spectra of independent GUE minor processes. The proof is based on new quantitative analysis of a formula to compute Schur functions at general points discovered in \cite{ZL18}.

Figures (4)

• Figure 2.1: A rail yard graph with LR sequence $\underline{a}=\{L,R,R,L\}$and sign sequence $\underline{b}=\{+,+,-,-\}$. Odd vertices are represented by red points, and even vertices are represented by blue points. Dark lines represent a pure dimer covering. Assume that above the horizontal line $y=4$, only horizontal edges with an odd vertex on the left are present in the dimer configuration; and below the horizontal line $y=-4$, only horizontal edges with an even vertex on the left are present in the dimer configuration. $l=0$ and $r=3$; red vertices have abscissas $x=-1,x=1,x=3,x=5,x=7$; blue vertices have abscissas $x=0,x=2,x=4,x=6$.Parameters atre given by $(a_0,b_0)=(L,+)$, $(a_1,b_1)=(R,+)$,$(a_2,b_2)=(R,-)$ and $(a_3,b_3)=(L,-)$.
• Figure 2.2: Young diagram corresponding to the partition $\lambda=(5,3,3,1)$.
• Figure 2.3: Particle-hole configuration corresponding to the dimer covering in Figure \ref{['fig:rye']} at red vertices. Particles are represented by red dots, while holes are represented by red circles. Each particle/hole represents the configuration of an odd vertex (represented by red dots) in the same location of Figure \ref{['fig:rye']}. The corresponding sequence of partitions (from the left to the right) is given by $\emptyset\prec(2,0,\ldots)\prec' (3,1,1,0\ldots)\succ'(2,0,\ldots)\succ \emptyset$.
• Figure 2.4: A rail yard graph with LR sequence $\underline{a}=\{L,R,L\}$and sign sequence $\underline{b}=\{+,+,-\}$. Odd vertices are represented by red points, and even vertices are represented by blue points. Dark lines represent a dimer covering with left boundary configuration given by a piecewise boundary condition and right boundary configuration given by an empty boundary condition.

Theorems & Definitions (47)

• Theorem 1.1
• Lemma 2.1: hciz
• Remark 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Lemma 2.7
• proof