论文标题
Schur的渐近学在几乎楼梯隔板上起作用
Asymptotics of Schur functions on almost staircase partitions
论文作者
论文摘要
我们研究几乎是楼梯的分区$λ$的Schur多项式的渐近学；更确切地说，与$（（M-1）（N-1），（N-1）（M-1）（N-2），\ ldots，（M-1），0）$不同的分区最多是一个组件，即$ n \ rightArrow \ rightarrow \ infty $，用于一个正Integer $ m \ ge 1 $ n $ n $ n $ n $。通过应用SCHUR函数的确定公式或积分表示，我们表明$ \ frac {1} {n} \ log \ frac {s_λ（u_1，u_1，\ ldots，\ ldots，u_k，u_k，x_ {k+1} $ k $单变量的全态函数，每个功能都取决于$ 1 \ leq i \ leq k $的可变$ u_i $，而当只有许多不同的$ x_i $的$ x_i $，每个$ u_i $且每个$ u_i $均在$ x_i $的附近，为$ x_i $，为$ n \ rightarrow \ rightarrow \ rightarrow \ iffty $。结果与大数字定律和中央限制定理有关，用于与某些边界条件合同的方形 - 甲状腺晶格上的二聚体配置。
We study the asymptotics of Schur polynomials with partitions $λ$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots,(m-1),0)$ by at most one component at the beginning as $N\rightarrow \infty$, for a positive integer $m\ge 1$ independent of $N$. By applying either determinant formulas or integral representations for Schur functions, we show that $\frac{1}{N}\log \frac{s_λ(u_1,\ldots,u_k, x_{k+1},\ldots,x_N)}{s_λ(x_1,\ldots,x_N)}$ converges to a sum of $k$ single-variable holomorphic functions, each of which depends on the variable $u_i$ for $1\leq i\leq k$, when there are only finitely many distinct $x_i$'s and each $u_i$ is in a neighborhood of $x_i$, as $N\rightarrow\infty$. The results are related to the law of large numbers and central limit theorem for the dimer configurations on contracting square-hexagon lattices with certain boundary conditions.
