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We analyze a symmetric Markov multinomial model simulating a Galton board, where a ball's trajectory is determined by directional choices at each peg with a fixed probability of maintaining direction versus random redirection. The research investigates the conditions under which the most likely bin on each face of the simplex becomes as probable as the corners, identifying a second-order tie-breaking mechanism based on explicit constants for each face. The key finding is that for a sufficiently long board, the most likely bin transitions from corners to the center as the expected number of turns grows logarithmically, with non-uniform starts or weak fields potentially favoring intermediate faces.
The most likely bin in a Galton board simulation unexpectedly jumps from corners to the center as the board lengthens, driven by a subtle second-order effect on directional choices.
We observe a ball bouncing down a Galton board with any number of directions. At each peg it either keeps its direction with some fixed probability or randomly turns to one of the other directions. Its bin measures how often it went each way, a point of a simplex. When the ball rarely turns, the most likely bins are the corners, which only a ball that never turns can reach. We ask when the best bin of each face of the simplex becomes as likely as a corner. To first order every face catches up at the same moment. We break this tie at second order, with an explicit constant for each face. Hence on a long board, as the expected number of turns grows to any fixed multiple of the length's logarithm, the most likely bin jumps once, from the corners straight to the center. A nonuniform start or a weak external field changes the constants, potentially allowing an intermediate face to win.