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We prove that the random functions stemming from two non-isomorphic words can be discriminated with probability arbitrarily close to $1$ with a bounded number of samples, when $n$ goes to infinity. Equivalently, the total variation distance between their distributions is bounded away from zero. The proof relies on the study of certain auto-correlation functions appearing in the variance of the weighted number of quasi-leaves.
Whether the total variation distance goes to 1, or equivalently, whether one sample is enough to distinguish the words with high probability, is the major question we leave open. We show that this is the case when the two words have different lengths or different exponents.
From: Guillaume Chapuy [view email]
[v1]
Mon, 30 Mar 2026 19:16:13 UTC (1,099 KB)
[v2]
Sun, 19 Jul 2026 14:47:13 UTC (982 KB)
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