I've heard the vague "it's 50%" before, but honestly that just raises more questions.
Y'all are the best people to ask, so I'd like to know if it's a flat % for every green dot, or for any distance traveled or what.
I don't know the exact specifics. It's possible it's listed in the BMG on GameFAQs.
But offhand: the odds are not increased the higher the number of dots traveled. The game runs a check for every dot you pass over and runs a second check for any dot you deliberately stop on - that second check being the key behind the Smart Encounters hack that disables the passing over check and sets the landed on check to have a 100% encounter rate.
Everything to know about random battles has been discovered.
I personally know all of it.
It's a flat 30%. What are your other questions?
If you want to check it out yourself, you could try to open the random battle editor and see for yourself, though I would guess there's a lot you wouldn't understand without asking some questions first.
You can find the random battle editor here:
https://ffhacktics.com/wiki/Tools#Random_Battle_Editor