Gibbs Measures with Multilinear Forms
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arXiv
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| Format: | Preprint |
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2023
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| author | Bhattacharya, Sohom Deb, Nabarun Mukherjee, Sumit |
| author_facet | Bhattacharya, Sohom Deb, Nabarun Mukherjee, Sumit |
| contents | In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain sufficient conditions for replica-symmetry, and provide examples to show why these conditions are also necessary. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the \enquote{local fields/magnetization}, the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14600 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gibbs Measures with Multilinear Forms Bhattacharya, Sohom Deb, Nabarun Mukherjee, Sumit Probability Mathematical Physics Combinatorics 82B20, 05C80 In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain sufficient conditions for replica-symmetry, and provide examples to show why these conditions are also necessary. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the \enquote{local fields/magnetization}, the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest. |
| title | Gibbs Measures with Multilinear Forms |
| topic | Probability Mathematical Physics Combinatorics 82B20, 05C80 |
| url | https://arxiv.org/abs/2307.14600 |