Send X across a channel Y = X + noise and ask what input is optimal.
An amplitude budget forces a discrete codebook; heavy-tailed (Lévy / α-stable) noise cuts two ways —
it resists discreteness in the channel but creates it (bimodality) for a particle relaxing in a steep well.
α = 2 (Gaussian)your αα = 1 (Cauchy)
Lighter α (smaller index) is simultaneously more peaked in the centre and fatter in the
tails (note the log scale). The characteristic function is φ(k) = exp(−|σ₀k|α) — the one object that
drives the channel, the information, and the diffusion in the other two tabs.
—
Red stems = the capacity-achieving input (Blahut–Arimoto) under |X| ≤ A. Watch the
staircase: as the budget grows the optimum adds mass points one at a time (2 → 3 → 4 …). Push α toward 1
(Cauchy) and the codebook needs a much larger budget before extra levels appear — heavy tails blur the channel and
cost capacity.
Gaussian (α=2)Lévy (your α)potential V(x)
t = 0.00
A particle in the steep well V(x) = (x/x₀)4 driven by α-stable noise obeys the
fractional Fokker–Planck equation ∂ₜp = ∂ₓ(p V′) − D(−Δ)α/2p. Gaussian noise relaxes to the usual
unimodal Boltzmann state; Lévy flights split it into a bimodal state — long jumps overshoot the centre and
pile up on the walls. This is where heavy tails genuinely manufacture discrete structure.