Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field
Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the condu...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Biskup, Marek [verfasserIn] |
|---|
| Format: |
Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
2019 |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
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| Übergeordnetes Werk: |
Enthalten in: Communications in mathematical physics - Springer Berlin Heidelberg, 1965, 373(2019), 1 vom: 26. Nov., Seite 45-106 |
|---|---|
| Übergeordnetes Werk: |
volume:373 ; year:2019 ; number:1 ; day:26 ; month:11 ; pages:45-106 |
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| DOI / URN: |
10.1007/s00220-019-03589-z |
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OLC2038921970 |
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| 520 | |a Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. | ||
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10.1007/s00220-019-03589-z doi (DE-627)OLC2038921970 (DE-He213)s00220-019-03589-z-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin (orcid)0000-0001-5560-6518 aut Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. Ding, Jian aut Goswami, Subhajit aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 373(2019), 1 vom: 26. Nov., Seite 45-106 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:373 year:2019 number:1 day:26 month:11 pages:45-106 https://doi.org/10.1007/s00220-019-03589-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4277 AR 373 2019 1 26 11 45-106 |
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10.1007/s00220-019-03589-z doi (DE-627)OLC2038921970 (DE-He213)s00220-019-03589-z-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin (orcid)0000-0001-5560-6518 aut Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. Ding, Jian aut Goswami, Subhajit aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 373(2019), 1 vom: 26. Nov., Seite 45-106 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:373 year:2019 number:1 day:26 month:11 pages:45-106 https://doi.org/10.1007/s00220-019-03589-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4277 AR 373 2019 1 26 11 45-106 |
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10.1007/s00220-019-03589-z doi (DE-627)OLC2038921970 (DE-He213)s00220-019-03589-z-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin (orcid)0000-0001-5560-6518 aut Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. Ding, Jian aut Goswami, Subhajit aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 373(2019), 1 vom: 26. Nov., Seite 45-106 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:373 year:2019 number:1 day:26 month:11 pages:45-106 https://doi.org/10.1007/s00220-019-03589-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4277 AR 373 2019 1 26 11 45-106 |
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10.1007/s00220-019-03589-z doi (DE-627)OLC2038921970 (DE-He213)s00220-019-03589-z-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin (orcid)0000-0001-5560-6518 aut Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. Ding, Jian aut Goswami, Subhajit aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 373(2019), 1 vom: 26. Nov., Seite 45-106 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:373 year:2019 number:1 day:26 month:11 pages:45-106 https://doi.org/10.1007/s00220-019-03589-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4277 AR 373 2019 1 26 11 45-106 |
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10.1007/s00220-019-03589-z doi (DE-627)OLC2038921970 (DE-He213)s00220-019-03589-z-p DE-627 ger DE-627 rakwb eng 530 510 VZ Biskup, Marek verfasserin (orcid)0000-0001-5560-6518 aut Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. Ding, Jian aut Goswami, Subhajit aut Enthalten in Communications in mathematical physics Springer Berlin Heidelberg, 1965 373(2019), 1 vom: 26. Nov., Seite 45-106 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:373 year:2019 number:1 day:26 month:11 pages:45-106 https://doi.org/10.1007/s00220-019-03589-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4277 AR 373 2019 1 26 11 45-106 |
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Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field |
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Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. © Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
| abstractGer |
Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. © Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
| abstract_unstemmed |
Abstract Given any $$\gamma {>}0$$ and for $$\eta {=}\{\eta _v\}_{v\in {\mathbb {Z}}^2}$$ denoting a sample of the two-dimensional discrete Gaussian free field on $${\mathbb {Z}}^2$$ pinned at the origin, we consider the random walk on $${\mathbb {Z}}^2$$ among random conductances where the conductance of edge (u, v) is given by $$\mathrm {e}^{\gamma (\eta _u + \eta _v)}$$. We show that, for almost every $$\eta $$, this random walk is recurrent and that, with probability tending to 1 as $$T\rightarrow \infty $$, the return probability at time 2T decays as $$T^{-1+o(1)}$$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius N scales as $$N^{\psi (\gamma )+o(1)}$$ with $$\psi (\gamma )>2$$ for all $$\gamma >0$$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance N behaves as $$N^{o(1)}$$. © Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
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| title_short |
Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field |
| url |
https://doi.org/10.1007/s00220-019-03589-z |
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| author2 |
Ding, Jian Goswami, Subhajit |
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| doi_str |
10.1007/s00220-019-03589-z |
| up_date |
2024-07-03T20:52:33.634Z |
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