What question did this study set out to answer?

This research aims to establish the relationship between strong and very strong disorder in directed polymers through their partition functions.

July 31, 2026Annales Scientifiques de l École Normale Supérieure3 citationsOpen Access

Strong disorder and very strong disorder are equivalent for directed polymers

SJStefan Junk HLHubert Lacoin

Key Points

This research aims to establish the relationship between strong and very strong disorder in directed polymers through their partition functions.
Analyzed the normalized partition function $W^β_n$ of directed polymers on $$ for varying values of inverse temperature $β$.
Proved the existence of a critical point $β_c$ differentiating weak from very strong disorder.
Incorporated a bounded above assumption on the environment to support the proof.
For $β eq β_c$, the normalized partition function converges exponentially fast to zero for $β o (β_c, \\infty)$, indicating very strong disorder.
A non-degenerate limit for $W^β_n$ exists for all $β eq[0, β_c]$, confirming weak disorder for this range.
Resolved a conjecture held for twenty years by Comets, Yoshida, Carmona, and Hu.

Abstract

We show that if the normalized partition function W^βₙ of the directed polymer model on Zᵈ converges to zero, then it does so exponentially fast. This implies that there exists a critical value βc for the inverse temperature such that the normalized partition function has a non-degenerate limit for all β 0, βc -- weak disorder holds -- while for β (βc, ) it converges exponentially fast to zero -- very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above.

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