The limit shape of convex lattice polygons
About 10 years ago Vershik asked whether a ``limit shape'' to some set of convex lattice polygons exists. This paper gives a partial answer to that question: the answer is yes for \({\mathcal P}_ n\), the set of all convex \(Z^ 2\)-lattice polygons lying in the square \([-n, n]^ 2\). For the precise statement of the theorem the set \({\mathcal P}_ n\) needs to be normalized: Let \[ {\mathcal P}_ n^ 0= \bigl\{ {\textstyle {1\over n}} P:\;P\in {\mathcal P}_ n \bigr\} \] and let \[ L= \bigl\{ (x,y)\in \mathbb{R}^ 2:\;\sqrt {1-| x|} \sqrt {1- | y|} \geq 1 \bigl\}; \] \(L\) is a convex set in \([-1, 1]^ 2\) bounded by four parabola-arcs each of them touching consecutive edges of the square at their midpoints. If \(P\in {\mathcal P}_ n^ 0\), let \(\chi_ P (x)\) be the characteristic function of \(P\). The main ``limit shape'' theorem concerning lattice polygons says that ``almost all'' polygons in \({\mathcal P}_ n^ 0\) are very close to \(L\): Theorem 1. The average of \(\chi_ P (x)\) over all \(P\in {\mathcal P}_ n^ 0\) tends to 1 if \(x\in \int L\) and to 0 if \(x\not\in L\). The author also states a similar result working with convex lattice curves (instead of polygons). Let \({\mathcal C}_ n\) be the set of all convex lattice curves connecting \((0,n)\) to \((n,0)\) within the triangle with vertices \((0,n)\), \((0,0)\), \((n,0)\) and let \({\mathcal C}_ n^ 0= \{ {1\over n} C\): \(C\in {\mathcal C}_ n\}\). Let \(\Gamma= \{(x,y)\in \mathbb{R}^ 2\): \(\sqrt {x} \sqrt {y}=1\}\). \(\Gamma\) is an arc of the parabola that touches the axes at \((1,0)\) and \((0,1)\). Let \(U_ \varepsilon (\Gamma)\) denote the \(\varepsilon\)-neighborhood of \(\Gamma\). Then Theorem 2. For every \(\varepsilon>0\) there is an \(n_ 0 (\varepsilon)\) such that for \(n>n_ 0 (\varepsilon)\) the number of \(C\in {\mathcal C}_ n^ 0\) not lying in \(U_ \varepsilon (\Gamma)\) is less than \(\varepsilon |{\mathcal C}_ n^ 0 |\). This paper presents also a summary of several interesting questions related with the ``limit shape'' theorems. Some of them have been solved by the author and will be published in subsequent papers, and the remaining questions are still open.
scientific article; zbMATH DE number 764227