Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$

Question

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators $$ L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\\0&1\\\end{pmatrix}. $$ Let $\SL_2^{(n)}(\mathbb{N})$ designate the set of words of length $n$ in these generators. I have a couple of questions about this family that seem within reach (i.e. are already known or could be determined by the good folks of mathoverflow).

1. What is the distribution of $\{\tr(M) : M\in \SL_2^{(n)}(\mathbb{N})\}$?
2. What is the distribution of $\{\|M\|_{\infty} : M\in \SL_2^{(n)}(\mathbb{N})\}$?
3. What is the joint distribution of the two quantities above?

I'd prefer to know the limiting distribution $n\to\infty$ and some useful error term (if they exist), but any non-trivial bounds are welcome, or approximations to relevant functions of the matrices as functions of the $\{L,R\}$ decomposition $$ M(w)=\prod_{i=0}^{n-1}G_{w_i}, \quad w\in\{0,1\}^n, \quad G_0=L, \quad G_1=R. $$

Here are a few pictures (of dubious quality); scatterplot on top of bar plot (could've just done a histogram but was more curious about some of the exact values).

Traces, $n=15,20,25$:

$L^{\infty}$, $n=15,20,25$:

Joint $n=15$ (the linear relationship seems to be $\tr(M)\approx \frac{9}{8}\|M\|_{\infty}$):

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EDIT: Thought I'd throw some gifs of distributions with changing $n$. Trace: Sup norm:

Answer 1

This problem is closely connected with number-theoretic model for spin chains. In this model, to a finite chain of spins each of which can be directed upwards ($\uparrow$) or downwards ($\downarrow$), a product of the matrices $$ A=\begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix},\quad B=\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} $$ is assigned, according to the rule ${\uparrow}=A$, ${\downarrow}=B$. For example, $$ {\uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow}= {\uparrow^3\downarrow^2\uparrow^4}=A^3B^2A^4. $$ The energy of a given configuration is $$ E(\uparrow^{a_1}\downarrow^{a_2}\uparrow^{a_3}\dotsm)= \log\left(\operatorname{Tr}(A^{a_1}B^{a_2}A^{a_3}\dotsm)\right). $$ Let $G$ be the free multiplicative monoid generated by the matrices $A$ and $B$. From a physical point of view it is interesting to know the asymptotic behaviour of the number of configurations $$ \Phi(N)=\bigl|\bigl\{C\in G: \operatorname{Tr} C=N \bigr\}\bigr|\qquad(N\ge 3) $$ with a given energy, and the number of configurations in which the energy does not exceed a given quantity, $$ \Psi(N)=\bigl |\bigl\{C\in G: 3\le\operatorname{Tr} C\leqslant N \bigr\}\bigr |= \sum_{3\le n\leqslant N}\Phi(n). $$ It is known, that $$ \Psi(N)=N^2(c_1\log N+c_0)+{O}(N^{3/2}\log^4N), $$ where $$ c_1=\frac{1}{\zeta(2)},\quad c_0= \frac{1}{\zeta(2)}\left(\gamma-\frac{3}{2}-\frac{\zeta'(2)} {\zeta(2)}\right). $$ And this is a special case of a more general result concerning the Gauss–Kuz’min statistics for spin chains and Gauss–Kuz’min statistics for the quadratic irrationals. For more details see the paper Spin chains and Arnold's problem on the Gauss–Kuz'min statistics for quadratic irrationals.

Answer 2

I believe that, as $n$ tends to infinity, if $w$ is a random word of length $n$ then $\frac{1}{n} \log(||w||)$ converges. The limit is called a Lyapunov exponent.

A bit of googling gives references - here is one of the first: Maximal Lyapunov exponents for random matrix products, by Mark Pollicott. In that paper, Pollicott restricts to the case where the generators of the semigroup are strictly positive matrices. He gives a concrete example for integral two-by-two matrices in his Example 0.1 (on page 211).