One step replica symmetry breaking and extreme order statistics of logarithmic REMs
Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted d...
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| Format: | Article |
| Language: | English |
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SciPost
2016-12-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.1.2.011 |
| _version_ | 1828182276149608448 |
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| author | Xiangyu Cao, Yan V. Fyodorov, Pierre Le Doussal |
| author_facet | Xiangyu Cao, Yan V. Fyodorov, Pierre Le Doussal |
| author_sort | Xiangyu Cao, Yan V. Fyodorov, Pierre Le Doussal |
| collection | DOAJ |
| description | Building upon the one-step replica symmetry breaking formalism, duly
understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ("infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$-noise. |
| first_indexed | 2024-04-12T06:14:46Z |
| format | Article |
| id | doaj.art-40824cab4ff344a68ceb3bed92a7c9a2 |
| institution | Directory Open Access Journal |
| issn | 2542-4653 |
| language | English |
| last_indexed | 2024-04-12T06:14:46Z |
| publishDate | 2016-12-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj.art-40824cab4ff344a68ceb3bed92a7c9a22022-12-22T03:44:31ZengSciPostSciPost Physics2542-46532016-12-011201110.21468/SciPostPhys.1.2.011One step replica symmetry breaking and extreme order statistics of logarithmic REMsXiangyu Cao, Yan V. Fyodorov, Pierre Le DoussalBuilding upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ("infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$-noise.https://scipost.org/SciPostPhys.1.2.011 |
| spellingShingle | Xiangyu Cao, Yan V. Fyodorov, Pierre Le Doussal One step replica symmetry breaking and extreme order statistics of logarithmic REMs SciPost Physics |
| title | One step replica symmetry breaking and extreme order statistics of logarithmic REMs |
| title_full | One step replica symmetry breaking and extreme order statistics of logarithmic REMs |
| title_fullStr | One step replica symmetry breaking and extreme order statistics of logarithmic REMs |
| title_full_unstemmed | One step replica symmetry breaking and extreme order statistics of logarithmic REMs |
| title_short | One step replica symmetry breaking and extreme order statistics of logarithmic REMs |
| title_sort | one step replica symmetry breaking and extreme order statistics of logarithmic rems |
| url | https://scipost.org/SciPostPhys.1.2.011 |
| work_keys_str_mv | AT xiangyucaoyanvfyodorovpierreledoussal onestepreplicasymmetrybreakingandextremeorderstatisticsoflogarithmicrems |