A central limit theorem for two-dimensional directed polymers with critical spatial correlation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915527252770816 |
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| author | Cosco, Clément Cottini, Francesca Donadini, Anna |
| author_facet | Cosco, Clément Cottini, Francesca Donadini, Anna |
| contents | On the 1+2 dimensional lattice, we consider a directed polymer in a random Gaussian environment that is independent in time and correlated in space. The spatial correlation is supposed to decay as $(\log |x|)^a /|x|^{2}$, $a>-1$, where the square in the polynomial is known to be critical (Lacoin, Ann. Prob. (2011)). We introduce an intermediate regime of temperature $β_N \propto \hat β/(\log N)^{\frac{a+2}{2}}$, under which the log-partition function $\log W_N^{β_N}$ converges in distribution towards a Gaussian random variable if $\hat β\in (0,\hat β_c)$, whereas $W_N^{β_N}$ vanishes for $\hat β\geq \hat β_c$. The variance of the limiting Gaussian distribution, which is given by an inverse Bessel function, is determined by an induction scheme whose multi-scale dependence reflects the critical nature of the correlation. The Gaussianity of the limit follows from a decoupling argument of Cosco, Donadini (2024+). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16694 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A central limit theorem for two-dimensional directed polymers with critical spatial correlation Cosco, Clément Cottini, Francesca Donadini, Anna Probability Primary: 82D60, Secondary: 82B44, 60F05 On the 1+2 dimensional lattice, we consider a directed polymer in a random Gaussian environment that is independent in time and correlated in space. The spatial correlation is supposed to decay as $(\log |x|)^a /|x|^{2}$, $a>-1$, where the square in the polynomial is known to be critical (Lacoin, Ann. Prob. (2011)). We introduce an intermediate regime of temperature $β_N \propto \hat β/(\log N)^{\frac{a+2}{2}}$, under which the log-partition function $\log W_N^{β_N}$ converges in distribution towards a Gaussian random variable if $\hat β\in (0,\hat β_c)$, whereas $W_N^{β_N}$ vanishes for $\hat β\geq \hat β_c$. The variance of the limiting Gaussian distribution, which is given by an inverse Bessel function, is determined by an induction scheme whose multi-scale dependence reflects the critical nature of the correlation. The Gaussianity of the limit follows from a decoupling argument of Cosco, Donadini (2024+). |
| title | A central limit theorem for two-dimensional directed polymers with critical spatial correlation |
| topic | Probability Primary: 82D60, Secondary: 82B44, 60F05 |
| url | https://arxiv.org/abs/2509.16694 |