The topology and geometry of random square-tiled surfaces

Abstract

A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. In this paper we consider a randomizing model for STSs and generalizations to branched covers of other simple translation surfaces which we call polygon-tiled surfaces. We obtain a local central limit theorem for the genus and subsequently obtain that the distribution of the genus is asymptotically normal. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. We show that asymptotically almost surely the set of holonomy vectors of a random STS contains the set of primitive vectors of ${\mathbb {Z}}^2$ and with probability approaching 1/e, these sets are equal.

Appendix A: Tools from combinatorics

Since our randomizing model is combinatorial, the overall strategy is to convert geometric questions into combinatorial ones about the model, and use combinatorial tools to answer them. Hence, in this section we provide a brief exposition on the relevant combinatorial notions and state the relevant combinatorial lemmas. Majority of the material in this section can be found in the textbook by Stanley [41, 42].

1.1 Partitions and the hook-length formula

A partition $n \in {\mathbb {N}}$ is a sequence $\lambda =(\lambda _1, \dots , \lambda _k) \in {\mathbb {N}}^k$ such that $\sum \lambda _i = n$ and $\lambda _1 \ge \lambda _2 \ge \dots \ge \lambda _k$. Two partitions are identical if they only differ in the number of zeros. For example $(3, 2, 1, 1) = (3, 2, 1, 1, 0, 0)$. Informally a partition can be thought of as a way of writing n as a sum $\lambda _1 + \dots + \lambda _k$ disregarding the order the $\lambda _i$ (since there is a canonical way of writing such a sum as a partition). If $\lambda$ is a partition of n, we write $\lambda \vdash n$. The non-zero $\lambda _i$ are called parts of $\lambda$ and we say that $\lambda$ has k parts if $k = \#\{i: \lambda _i > 0\}$. If $\lambda$ has $\xi _i$ parts equal to i, then we can write $\lambda = \langle 1^{\xi _1}, 2^{\xi _2}, \dots \rangle$ where terms with $\xi _i = 0$ and the superscript $\xi _i = 1$ is omitted. Denote by $\hbox \mathrm{Par}(n)$, the set of all partitions of n and set $\hbox \mathrm{Par}= \bigcup _{n \in {\mathbb {N}}} \hbox \mathrm{Par}(n)$.

For a partition $(\lambda _1, \dots , \lambda _k) \vdash n$, we can draw a left-justified array of boxes with $\lambda _i$ boxes in the i-th row. This is called the Young diagram associated to partition $(\lambda _1, \dots , \lambda _k)$. The squares in a Young diagram can be identified using tuples (i, j) where i is the row corresponding to part $\lambda _i$ and $1 \le j\le \lambda _i$ is the position of the square along that row. Given a square $r = (i,j) \in \lambda$, define the hook length of $\lambda$ at r as the number of squares directly to the right or directly below r, counting r itself once. The hook length product of $\lambda$, denoted $H_\lambda$ is the product,

$$\begin{aligned} H_\lambda = \prod _{u \in \lambda } h(u) \end{aligned}$$

Likewise, define the content $\hbox \mathrm{cont}(r)$ of $\lambda$ at $r = (i, j)$ by $\hbox \mathrm{cont}(r) = j-i$. In general we obtain a Young tableau by filling in the boxes of the Young diagram with entries from a totally ordered set (usually a set of positive integers). It is called a semistandard Young tableau (SSYT) if the entries of the tableau weakly increase along each row and strictly increase down each column. The type of an SSYT is a sequence $\xi = (\xi _1, \dots )$ where the SSYT contains $\xi _1 1's$, $\xi _2 2's$ and so on. See Fig. 8 for examples of Young diagram and Young tableau.

Fig. 8

Young diagram and tableau for the partition (5,3,3,2,1). From left to right: Young diagram; Tableau with hook lengths; Tableau with contents; Example of an SSYT

Partitions are particularly relevant to us, since the cycle type of a permutation $\pi \in S_n$ is a partition of n, and two permutations are conjugate if and only if they have the same cycle type. Hence, partitions index conjugacy classes of $S_n$, and we will denote $\hbox \mathrm{cyc}(\pi )$ to be the partition obtained from the cycle type of $\pi$. Moreover, partitions of n also index the irreducible representations and irreducible characters of $S_n$ canonically. We will denote by $\rho ^\lambda$ and $\chi ^\lambda$ the irreducible representation and character associated indexed by the partition $\lambda$. One instance of the deep relation between partitions and the representation theory of the symmetric group is the Hook-length formula which gives a relation between the degree (or dimension) of a representation and the hook length product of the partition that is used to index it:

Lemma A.1

(Hook-length formula) For a partition $\lambda \vdash n$, and associated representation $\rho ^\lambda \in {\hat{S}}_n$, the degree is given by

$$\begin{aligned} \dim (\rho ^\lambda ) = \frac{n!}{H_\lambda } \end{aligned}$$

1.2 Symmetric functions and the Murnaghan–Nakayama rule

Let $x = (x_1, x_2,\dots )$ be a set of indeterminates, and let $n \in {\mathbb {N}}$.

Denote the set of all homogeneous symmetric functions of degree n over ${\mathbb {Q}}$ as $\Lambda ^n$. Note that $\Lambda ^n$ is a ${\mathbb {Q}}$-vector space with many different standard bases, two of which we will utilize. The first of which are called power sum symmetric functions, denoted $p_\lambda$ and indexed by partitions $\lambda$. They are defined as,

$$\begin{aligned}&p_n := \sum _{i} x_i^n, n \ge 1 \text { (with }p_0 = 1)\\&p_\lambda := p_{\lambda _1}p_{\lambda _2}\dots \text { if }\lambda = (\lambda _1, \lambda _2, \dots , ) \end{aligned}$$

The second set of symmetric functions we will need are called Schur functions, denoted $s_\lambda$ and also indexed by partitions $\lambda$. They are defined as the formal power series,

$$\begin{aligned} s_\lambda (x) = \sum _{T} x^T \end{aligned}$$

where

• the sum is over all SSYTs T of shape $\lambda$.

• $x^T = x_1^{\xi _1(T)} x_2^{\xi _2(T)} \dots$ if T is an SSYT of type $\xi$.

While it is not hard to see that power sum symmetric functions are indeed symmetric functions, it is a non-trivial theorem that Schur functions are indeed symmetric functions. There is a deep connection between Schur functions, power sum symmetric functions and irreducible characters of $S_n$. This relation is called the Murnaghan-Nakayama rule:

Theorem A.2

(Murnaghan-Nakayama Rule) For $\lambda \vdash n$,

$$\begin{aligned} s_\lambda = \sum _{\nu \vdash n} z^{-1}_\nu \chi ^\lambda (\nu )p_\nu \end{aligned}$$

where $\chi ^\lambda$ is the irreducible character of $S_n$ indexed by $\lambda$ and $z^{-1}_\nu = \frac{\#\{\sigma \in S_n: \hbox \mathrm{cyc}(\sigma ) = \nu \}}{n!}$

Conversely, the power sum symmetric functions can be transformed into the Schur functions as follows:

Theorem A.3

The power sum symmetric functions can be expressed as a linear combination of the Schur functions in the following way:

$$\begin{aligned} p_\mu = \sum _{\lambda \vdash n} \chi ^\lambda (\mu ) s_\lambda \end{aligned}$$

Next, let $\hbox \mathrm{CF}^n$ be the set of class functions $f: S_n \rightarrow {\mathbb {Q}}$. Then, there exists a natural inner product on $\hbox \mathrm{CF}^n$ given by,

$$\begin{aligned} \langle f, g\rangle = \frac{1}{n!}\sum _{\sigma \in S_n} f(\sigma )g(\sigma ) = \sum _{\lambda \vdash n} z^{-1}_\lambda f(\lambda )g(\lambda ) \end{aligned}$$

where $f(\lambda )$ denotes the value of f on the conjugacy class associated to partition $\lambda$. To each class function $f \in \hbox \mathrm{CF}^n$, one can associate a symmetric function of degree n via the linear transformation $\hbox \mathrm{ch}: \hbox \mathrm{CF}^n \rightarrow \Lambda ^n$, called the Frobenius characteristic map given by,

$$\begin{aligned} \hbox \mathrm{ch}f = \frac{1}{n!}\sum _{\sigma \in S_n} f(\sigma )p_{\hbox \mathrm{cyc}(\sigma )} = \sum _{\lambda \vdash n} z^{-1}_\lambda f(\lambda ) p_\lambda . \end{aligned}$$

(6)

Using Theorem A.3, the characteristic function is expressed as,

$$\begin{aligned} \hbox \mathrm{ch}f = \sum _{\lambda \vdash n} \langle f, \chi ^\lambda \rangle s_\lambda \end{aligned}$$

(7)

Finally, we state a particular specialization of $s_\lambda$ which we will use. This is Corollary 7.21.4 of [42]

Lemma A.4

For any $\lambda \in \hbox \mathrm{Par}$ and m a positive integer, we have

$$\begin{aligned} s_\lambda (1^m) = \prod _{r \in \lambda } \frac{m+\hbox \mathrm{cont}(r)}{h(r)} \end{aligned}$$

where $s_\lambda (1^m)$ means evaluating $s_\lambda$ by setting $x_1 = \dots x_m = 1$ and $x_i = 0$ for all $i > m$.