Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field
Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attent...
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Gespeichert in:
| Autor*in: |
Biskup, Marek [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2016 |
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| Übergeordnetes Werk: |
Enthalten in: Communications in mathematical physics - Springer Berlin Heidelberg, 1965, 345(2016), 1 vom: 29. Jan., Seite 271-304 |
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| Übergeordnetes Werk: |
volume:345 ; year:2016 ; number:1 ; day:29 ; month:01 ; pages:271-304 |
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| DOI / URN: |
10.1007/s00220-015-2565-8 |
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OLC2038913080 |
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| 520 | |a Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. | ||
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10.1007/s00220-015-2565-8 doi (DE-627)OLC2038913080 (DE-He213)s00220-015-2565-8-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin aut Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. Brownian Motion Extreme Point Point Process Random Measure Gibbs Measure Louidor, Oren aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 345(2016), 1 vom: 29. Jan., Seite 271-304 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:345 year:2016 number:1 day:29 month:01 pages:271-304 https://doi.org/10.1007/s00220-015-2565-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 345 2016 1 29 01 271-304 |
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10.1007/s00220-015-2565-8 doi (DE-627)OLC2038913080 (DE-He213)s00220-015-2565-8-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin aut Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. Brownian Motion Extreme Point Point Process Random Measure Gibbs Measure Louidor, Oren aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 345(2016), 1 vom: 29. Jan., Seite 271-304 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:345 year:2016 number:1 day:29 month:01 pages:271-304 https://doi.org/10.1007/s00220-015-2565-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 345 2016 1 29 01 271-304 |
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10.1007/s00220-015-2565-8 doi (DE-627)OLC2038913080 (DE-He213)s00220-015-2565-8-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin aut Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. Brownian Motion Extreme Point Point Process Random Measure Gibbs Measure Louidor, Oren aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 345(2016), 1 vom: 29. Jan., Seite 271-304 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:345 year:2016 number:1 day:29 month:01 pages:271-304 https://doi.org/10.1007/s00220-015-2565-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 345 2016 1 29 01 271-304 |
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10.1007/s00220-015-2565-8 doi (DE-627)OLC2038913080 (DE-He213)s00220-015-2565-8-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin aut Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. Brownian Motion Extreme Point Point Process Random Measure Gibbs Measure Louidor, Oren aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 345(2016), 1 vom: 29. Jan., Seite 271-304 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:345 year:2016 number:1 day:29 month:01 pages:271-304 https://doi.org/10.1007/s00220-015-2565-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 345 2016 1 29 01 271-304 |
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10.1007/s00220-015-2565-8 doi (DE-627)OLC2038913080 (DE-He213)s00220-015-2565-8-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin aut Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2016 Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. Brownian Motion Extreme Point Point Process Random Measure Gibbs Measure Louidor, Oren aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 345(2016), 1 vom: 29. Jan., Seite 271-304 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:345 year:2016 number:1 day:29 month:01 pages:271-304 https://doi.org/10.1007/s00220-015-2565-8 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 345 2016 1 29 01 271-304 |
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extreme local extrema of two-dimensional discrete gaussian free field |
| title_auth |
Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field |
| abstract |
Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. © The Author(s) 2016 |
| abstractGer |
Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. © The Author(s) 2016 |
| abstract_unstemmed |
Abstract We consider the discrete Gaussian Free Field in a square box in $${\mathbb{Z}^2}$$ of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as $${N \to \infty}$$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an rN-neighborhood thereof, we prove that the associated process tends, whenever $${r_N \to \infty}$$ and $${r_N/N \to 0}$$, to a Poisson point process with intensity measure $${Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}$$, where $${\alpha:= 2/\sqrt{g}}$$ with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field. © The Author(s) 2016 |
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| title_short |
Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field |
| url |
https://doi.org/10.1007/s00220-015-2565-8 |
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Louidor, Oren |
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| up_date |
2024-07-03T20:50:17.418Z |
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