Suppose $F_m := F[x_1, … , x_m]$ is a free group on $m$ generators $x_1, … , x_m$ and lets define Cayley ball $B_m^n := \{e, x_1, x_1^{-1}, … , x_m, x_m^{-1}\}^n$ as the set of all elements with Cayley length $n$ or less.

Suppose $R_1, … , R_l$ are $l$ random elements chosen uniformly from $B_m^n$. Then we can define a random group as $G(m, l, n) := \frac{F_m}{\langle \langle \{R_1, … , R_l\} \rangle \rangle}$.

Now will suppose that $m$ is fixed and $l = l(n)$ depends on $n$. We say that the random group $G(m, l, n)$ belongs to a class of groups $\mathfrak{U}$ almost surely iff $\lim_{n \to \infty} P(G(m, l, n) \in \mathfrak{U}) = 1$.

A following theorem was proved by Ollivier:

If $\lim_{n \to \infty} \frac{\ln(l(n))}{n(\ln(2m - 1))} > \frac{1}{2}$ then $G(m, l, n)$ is almost surely finite.

If $\lim_{n \to \infty} \frac{\ln(l(n))}{n(\ln(2m - 1))} < \frac{1}{2}$ then $G(m, l, n)$ is almost surely infinite

My question is:

Is there some sort of exact expression for limit probability $\lim_{n \to \infty} P(G(m, l(n), n) \text{ is finite})$ for arbitrary non-decreasing $l: \mathbb{N} \to \mathbb{N}$?

I, personally think, that it is very likely to be of the form

$$\lim_{n \to \infty} P(G(m, l(n), n) \text{ is finite}) = \lim_{n \to \infty} a^{-b^{\ln(2m - 1)n - 2\ln(l(n))}}$$

for some positive real numbers $a(m)$ and $b(m)$.

However, I do not know that for sure. That is just intuition mostly based on analogies with a somewhat similar "phase transition theorem" from a completely different field, that states:

Suppose $G(n, p(n))$ is an Erdos-Renyi random graph with $n$ vertices and edge probability $p(n)$. Then $\lim_{n \to \infty} P(G(n, p(n)) \text{ is connected}) = \lim_{n \to \infty} e^{-e^{\ln(n) - np(n)}}$

  • $\begingroup$ This seems more suited to MathOverflow than here, especially given that it's not been answered in over a year. $\endgroup$
    – user1729
    Jul 6, 2021 at 21:09


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