For disordered pinning models with non-degenerate disorder, conditioning the contact density between 0 and 1 yields largest gaps of order log n, so the pure-model big jump phenomenon disappears.
Billingsley, Probability and measure, 3rd edn (John Wiley & Sons, New York, 1995)
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Pinning models with contact number constraint: the effect of disorder
For disordered pinning models with non-degenerate disorder, conditioning the contact density between 0 and 1 yields largest gaps of order log n, so the pure-model big jump phenomenon disappears.