Sharp behavior of the free energy for the two-dimensional directed polymer model
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911730906431488 |
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| author | Berger, Quentin Nakajima, Shuta |
| author_facet | Berger, Quentin Nakajima, Shuta |
| contents | We consider the directed polymer model on $\mathbb{Z}^d$, in an i.i.d.\ random environment $ω=(ω_{n,x})_{n\geq 0,x\in\mathbb Z^d}$, focusing on the critical dimension $d=2$. Our main contribution is to give a sharp lower bound on the free energy in the high-temperature regime. Our proof uses a percolation argument inspired by Lacoin (2010), for which we introduce a key property of bounded ``$\log$-energy'': this property quantifies the regularity of the polymer measures at diffusive scales and we show that it propagates along open paths. Writing $\mathfrak{f}(β)$ for the quenched free energy, and setting $λ(β):=\log \mathbb E[e^{βω_{1,0}}]$ and $σ(β)^2:=e^{λ(2β)-2λ(β)}-1$, our lower bound combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025) gives
$$
-\mathfrak{f}(β) \asymp \exp{\Big(- \fracπ{σ^2(β)}\Big)},\quad \text{ as $β\downarrow 0$.}
$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30707 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp behavior of the free energy for the two-dimensional directed polymer model Berger, Quentin Nakajima, Shuta Probability Mathematical Physics Primary 60K35, 82B44, Secondary 82D60 We consider the directed polymer model on $\mathbb{Z}^d$, in an i.i.d.\ random environment $ω=(ω_{n,x})_{n\geq 0,x\in\mathbb Z^d}$, focusing on the critical dimension $d=2$. Our main contribution is to give a sharp lower bound on the free energy in the high-temperature regime. Our proof uses a percolation argument inspired by Lacoin (2010), for which we introduce a key property of bounded ``$\log$-energy'': this property quantifies the regularity of the polymer measures at diffusive scales and we show that it propagates along open paths. Writing $\mathfrak{f}(β)$ for the quenched free energy, and setting $λ(β):=\log \mathbb E[e^{βω_{1,0}}]$ and $σ(β)^2:=e^{λ(2β)-2λ(β)}-1$, our lower bound combined with Theorem 2.8 of Berger, Caravenna, and Turchi (2025) gives $$ -\mathfrak{f}(β) \asymp \exp{\Big(- \fracπ{σ^2(β)}\Big)},\quad \text{ as $β\downarrow 0$.} $$ |
| title | Sharp behavior of the free energy for the two-dimensional directed polymer model |
| topic | Probability Mathematical Physics Primary 60K35, 82B44, Secondary 82D60 |
| url | https://arxiv.org/abs/2605.30707 |