Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also...
| Main Authors: | , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Published: |
Institute of Mathematical Statistics
2018
|
| Online Access: | http://hdl.handle.net/1721.1/115337 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1811081772860964864 |
|---|---|
| author | Duplantier, Bertrand Rhodes, Rémi Vargas, Vincent Sheffield, Scott Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Duplantier, Bertrand Rhodes, Rémi Vargas, Vincent Sheffield, Scott Roger |
| author_sort | Duplantier, Bertrand |
| collection | MIT |
| description | In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables. |
| first_indexed | 2024-09-23T11:52:14Z |
| format | Article |
| id | mit-1721.1/115337 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T11:52:14Z |
| publishDate | 2018 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1153372022-09-27T22:31:12Z Critical Gaussian multiplicative chaos: Convergence of the derivative martingale Duplantier, Bertrand Rhodes, Rémi Vargas, Vincent Sheffield, Scott Roger Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables. National Science Foundation (U.S.) (Grant DMS-06-4558) 2018-05-11T18:34:54Z 2018-05-11T18:34:54Z 2014-09 2013-09 2018-05-10T16:52:18Z Article http://purl.org/eprint/type/JournalArticle 0091-1798 http://hdl.handle.net/1721.1/115337 Duplantier, Bertrand et al. “Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale.” The Annals of Probability 42, 5 (September 2014): 1769–1808 © 2014 Institute of Mathematical Statistics https://orcid.org/0000-0002-5951-4933 http://dx.doi.org/10.1214/13-AOP890 Annals of Probability Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Duplantier, Bertrand Rhodes, Rémi Vargas, Vincent Sheffield, Scott Roger Critical Gaussian multiplicative chaos: Convergence of the derivative martingale |
| title | Critical Gaussian multiplicative chaos: Convergence of the derivative martingale |
| title_full | Critical Gaussian multiplicative chaos: Convergence of the derivative martingale |
| title_fullStr | Critical Gaussian multiplicative chaos: Convergence of the derivative martingale |
| title_full_unstemmed | Critical Gaussian multiplicative chaos: Convergence of the derivative martingale |
| title_short | Critical Gaussian multiplicative chaos: Convergence of the derivative martingale |
| title_sort | critical gaussian multiplicative chaos convergence of the derivative martingale |
| url | http://hdl.handle.net/1721.1/115337 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT duplantierbertrand criticalgaussianmultiplicativechaosconvergenceofthederivativemartingale AT rhodesremi criticalgaussianmultiplicativechaosconvergenceofthederivativemartingale AT vargasvincent criticalgaussianmultiplicativechaosconvergenceofthederivativemartingale AT sheffieldscottroger criticalgaussianmultiplicativechaosconvergenceofthederivativemartingale |