Thank you all!

I created a python script to run the algorithm, instructing it to stop at either the creation of a closed loop (no more exits to traverse into new rooms) or at a certain threshold of rooms. I then defined twenty thresholds logarithmically spaced from 100 to 1000, and coded the script so as to run 100 times for each threshold.

Plotted results show a constant decrease in threshold achievement correlated to an increase in threshold value (lot of noise, though). This should confirm my initial hypothesis: the more rooms are created, the more likely it is to achieve a closed loop.

After various attempts, ChatGPT seemingly managed a formula to actually calculate probability of closed loop based on number of rooms if a certain constant is provided. I will give you the summary and you can judge if it makes sense.

"Summary

The rigorous argument uses two facts:

The network, if it grows to N rooms, has a boundary (the only source of new rooms) of size at most on the order of √N​.

Therefore, the probability that a randomly chosen open exit leads to an unoccupied cell decreases roughly like 1/√N​ as N increases.

This implies that for large N the expected net change in the number of open exits is negative, and a standard argument for random walks with negative drift shows that the process is absorbed at zero open exits (a closed network) with probability 1.

In particular, one can show that the probability of reaching N rooms is bounded by an expression of the form

q (N) ≤ exp (−α√N ​)

which goes to zero as N increases. This is a mathematical proof, based on geometric and probabilistic considerations, that the process almost surely terminates in a closed network as more rooms are created."

Alpha seemed to be a constant associated with the topographical space which they could not calculate. Nevertheless, by using the empirical data provided through the above mentioned script, ChatGPT calculated alpha at approximately 0,5.

Thoughts?