Extremes of the discrete Gaussian free field in dimension two
Extremes of the discrete Gaussian free field in dimension two
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DissertationPrüfungsdatum
28.01.2021Datum der Veröffentlichung
26.03.2021Erstgutachter
Bovier, AntonZweitgutachter
Kistler, NicolaMetadaten
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Inhalt
In recent years, there have been many advances towards an understanding of the extreme value theory of log-correlated random fields. Log-correlated random fields are conjectured to compose a universality class for the extremal values of strongly correlated fields. In the general context of extreme value statistics there are two natural basic questions to answer. Akin to the central limit theorem one may ask: Is there a deterministic recentring and rescaling such that the maximum value of the sequence converges to a non-trivial limit?
And second, if such a recentring and rescaling exists, how does the process look like when recentring and rescaling each random variable as done for the maximum value?
Both questions were answered in the context of independent identically distributed random variables during the first half of the past century. The theory developed in this context is commonly referred to as classical extreme value theory. We state the main results in the general case of independent identically distributed random variables and then turn to the case of Gaussian distributions.
To analyze the extreme value statistics of correlated models, it is natural to start with simple models that capture the essential details, which in our case are the hierarchical ones. We start with a rather classical model, the generalized random energy model (GREM), which can be realized as a branching random walk with Gaussian increments, and then discuss (variable-speed) branching Brownian motion (BBM), a model that has attracted a lot of interest in the last decade.
An important example of a log-correlated Gaussian random field is the two-dimensional discrete Gaussian free field (2d DGFF). It is a natural object of major interest both in mathematics and physics. Its extremal values have been investigated in the last 20 years.
We then introduce the model we studied, which is a generalization of the 2d DGFF, the so-called scale-inhomogeneous two-dimensional discrete Gaussian free field. Similarly to variable-speed BBM in the context of BBM, it allows for a richer class of correlation structures. It turns out that it is possible to classify its extremal values into three possible cases, one being the two-dimensional discrete Gaussian free field. In this thesis, we present our contributions in the study of the extremal values of the scale-inhomogeneous 2d DGFF. In any of the three possible cases and when there are only finitely many scales we determine the sub-leading order correction to the maximum value and prove tightness of the centred maximum. Moreover, in the case of weak correlations we provide a complete characterization of the extreme value theory of the scale-inhomogeneous 2d DGFF.
And second, if such a recentring and rescaling exists, how does the process look like when recentring and rescaling each random variable as done for the maximum value?
Both questions were answered in the context of independent identically distributed random variables during the first half of the past century. The theory developed in this context is commonly referred to as classical extreme value theory. We state the main results in the general case of independent identically distributed random variables and then turn to the case of Gaussian distributions.
To analyze the extreme value statistics of correlated models, it is natural to start with simple models that capture the essential details, which in our case are the hierarchical ones. We start with a rather classical model, the generalized random energy model (GREM), which can be realized as a branching random walk with Gaussian increments, and then discuss (variable-speed) branching Brownian motion (BBM), a model that has attracted a lot of interest in the last decade.
An important example of a log-correlated Gaussian random field is the two-dimensional discrete Gaussian free field (2d DGFF). It is a natural object of major interest both in mathematics and physics. Its extremal values have been investigated in the last 20 years.
We then introduce the model we studied, which is a generalization of the 2d DGFF, the so-called scale-inhomogeneous two-dimensional discrete Gaussian free field. Similarly to variable-speed BBM in the context of BBM, it allows for a richer class of correlation structures. It turns out that it is possible to classify its extremal values into three possible cases, one being the two-dimensional discrete Gaussian free field. In this thesis, we present our contributions in the study of the extremal values of the scale-inhomogeneous 2d DGFF. In any of the three possible cases and when there are only finitely many scales we determine the sub-leading order correction to the maximum value and prove tightness of the centred maximum. Moreover, in the case of weak correlations we provide a complete characterization of the extreme value theory of the scale-inhomogeneous 2d DGFF.
Schlagwörter
extremal process, discrete Gaussian free field, inhomogeneous environment, Gaussian processes, cluster process, Gaussian comparison
Klassifikation (DDC)
510 MathematikFels, Christian Joachim Maximilian: Extremes of the discrete Gaussian free field in dimension two. - Bonn, 2021. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-61573
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-61573
@phdthesis{handle:20.500.11811/9009,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-61573,
author = {{Christian Joachim Maximilian Fels}},
title = {Extremes of the discrete Gaussian free field in dimension two},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2021,
month = mar,
note = {In recent years, there have been many advances towards an understanding of the extreme value theory of log-correlated random fields. Log-correlated random fields are conjectured to compose a universality class for the extremal values of strongly correlated fields. In the general context of extreme value statistics there are two natural basic questions to answer. Akin to the central limit theorem one may ask: Is there a deterministic recentring and rescaling such that the maximum value of the sequence converges to a non-trivial limit?
And second, if such a recentring and rescaling exists, how does the process look like when recentring and rescaling each random variable as done for the maximum value?
Both questions were answered in the context of independent identically distributed random variables during the first half of the past century. The theory developed in this context is commonly referred to as classical extreme value theory. We state the main results in the general case of independent identically distributed random variables and then turn to the case of Gaussian distributions.
To analyze the extreme value statistics of correlated models, it is natural to start with simple models that capture the essential details, which in our case are the hierarchical ones. We start with a rather classical model, the generalized random energy model (GREM), which can be realized as a branching random walk with Gaussian increments, and then discuss (variable-speed) branching Brownian motion (BBM), a model that has attracted a lot of interest in the last decade.
An important example of a log-correlated Gaussian random field is the two-dimensional discrete Gaussian free field (2d DGFF). It is a natural object of major interest both in mathematics and physics. Its extremal values have been investigated in the last 20 years.
We then introduce the model we studied, which is a generalization of the 2d DGFF, the so-called scale-inhomogeneous two-dimensional discrete Gaussian free field. Similarly to variable-speed BBM in the context of BBM, it allows for a richer class of correlation structures. It turns out that it is possible to classify its extremal values into three possible cases, one being the two-dimensional discrete Gaussian free field. In this thesis, we present our contributions in the study of the extremal values of the scale-inhomogeneous 2d DGFF. In any of the three possible cases and when there are only finitely many scales we determine the sub-leading order correction to the maximum value and prove tightness of the centred maximum. Moreover, in the case of weak correlations we provide a complete characterization of the extreme value theory of the scale-inhomogeneous 2d DGFF.},
url = {https://hdl.handle.net/20.500.11811/9009}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-61573,
author = {{Christian Joachim Maximilian Fels}},
title = {Extremes of the discrete Gaussian free field in dimension two},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2021,
month = mar,
note = {In recent years, there have been many advances towards an understanding of the extreme value theory of log-correlated random fields. Log-correlated random fields are conjectured to compose a universality class for the extremal values of strongly correlated fields. In the general context of extreme value statistics there are two natural basic questions to answer. Akin to the central limit theorem one may ask: Is there a deterministic recentring and rescaling such that the maximum value of the sequence converges to a non-trivial limit?
And second, if such a recentring and rescaling exists, how does the process look like when recentring and rescaling each random variable as done for the maximum value?
Both questions were answered in the context of independent identically distributed random variables during the first half of the past century. The theory developed in this context is commonly referred to as classical extreme value theory. We state the main results in the general case of independent identically distributed random variables and then turn to the case of Gaussian distributions.
To analyze the extreme value statistics of correlated models, it is natural to start with simple models that capture the essential details, which in our case are the hierarchical ones. We start with a rather classical model, the generalized random energy model (GREM), which can be realized as a branching random walk with Gaussian increments, and then discuss (variable-speed) branching Brownian motion (BBM), a model that has attracted a lot of interest in the last decade.
An important example of a log-correlated Gaussian random field is the two-dimensional discrete Gaussian free field (2d DGFF). It is a natural object of major interest both in mathematics and physics. Its extremal values have been investigated in the last 20 years.
We then introduce the model we studied, which is a generalization of the 2d DGFF, the so-called scale-inhomogeneous two-dimensional discrete Gaussian free field. Similarly to variable-speed BBM in the context of BBM, it allows for a richer class of correlation structures. It turns out that it is possible to classify its extremal values into three possible cases, one being the two-dimensional discrete Gaussian free field. In this thesis, we present our contributions in the study of the extremal values of the scale-inhomogeneous 2d DGFF. In any of the three possible cases and when there are only finitely many scales we determine the sub-leading order correction to the maximum value and prove tightness of the centred maximum. Moreover, in the case of weak correlations we provide a complete characterization of the extreme value theory of the scale-inhomogeneous 2d DGFF.},
url = {https://hdl.handle.net/20.500.11811/9009}
}
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