Parabolic Anderson Model in Hyperbolic Spaces and Phase Transition Article Swipe

PDF

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2507.05530

Consider a Parabolic Anderson model (PAM) with Gaussian noise that is white in time and colored in space, where the spatial correlation decays polynomially with order $α$. In Euclidean spaces with dimension greater than $2$, it is well-understood that the critical value for $α$ is $2$. Specifically, for $α<2$, the second moment of the solution grows exponentially over time, while for $α>2$, there is a phase transition, from the second moment being uniformly bounded in time to exhibiting exponential growth in time when the inverse temperature increases. This critical behavior arises from the fact that in Euclidean space, Brownian motion tends to infinity at a speed of $\sqrt{t}$. The present work explores the PAM on a hyperbolic space. Given that Brownian motion in a hyperbolic space travels at a speed of $t$, one expects that $α=1$ would be the critical value for the above phenomena. We confirm that this intuition is indeed correct. Furthermore, we uncover a novel phase for $α<1$ in which the second moment explodes sub-exponentially, distinct from the behavior observed in Euclidean space.

Related Topics

Harry Anderson

America's Next Top Model

Beechcraft Model 18

Remington Model 700

Kyle Anderson (Basketball)

Mathematical Formulation Of The Standard Model

Gerald Anderson

Kevin Anderson (Tennis)

Lynn Anderson

Brock Anderson

Kristen Anderson-Lopez

Michael J. Anderson

Concepts

No concepts available.

Metadata

Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2507.05530
PDF: https://arxiv.org/pdf/2507.05530
OA Status: green
OpenAlex ID: https://openalex.org/W4416060369

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W4416060369

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.48550/arxiv.2507.05530

Digital Object Identifier
Title: Parabolic Anderson Model in Hyperbolic Spaces and Phase Transition

Work title
Type: preprint

OpenAlex work type
Language: en

Primary language
Publication year: 2025

Year of publication
Publication date: 2025-07-07

Full publication date if available
Authors: Xiaoping Geng, Cheng Ouyang

List of authors in order
Landing page: https://arxiv.org/abs/2507.05530

Publisher landing page
PDF URL: https://arxiv.org/pdf/2507.05530

Direct link to full text PDF
Open access: Yes

Whether a free full text is available
OA status: green

Open access status per OpenAlex
OA URL: https://arxiv.org/pdf/2507.05530

Direct OA link when available
Cited by: 0

Total citation count in OpenAlex

Full payload

id	https://openalex.org/W4416060369
doi	https://doi.org/10.48550/arxiv.2507.05530
ids.doi	https://doi.org/10.48550/arxiv.2507.05530
ids.openalex	https://openalex.org/W4416060369
fwci
type	preprint
title	Parabolic Anderson Model in Hyperbolic Spaces and Phase Transition
biblio.issue
biblio.volume
biblio.last_page
biblio.first_page
is_xpac	False
apc_list
apc_paid
language	en
locations[0].id	pmh:oai:arXiv.org:2507.05530
locations[0].is_oa	True
locations[0].source.id	https://openalex.org/S4306400194
locations[0].source.issn
locations[0].source.type	repository
locations[0].source.is_oa	True
locations[0].source.issn_l
locations[0].source.is_core	False
locations[0].source.is_in_doaj	False
locations[0].source.display_name	arXiv (Cornell University)
locations[0].source.host_organization	https://openalex.org/I205783295
locations[0].source.host_organization_name	Cornell University
locations[0].source.host_organization_lineage	https://openalex.org/I205783295
locations[0].license
locations[0].pdf_url	https://arxiv.org/pdf/2507.05530
locations[0].version	submittedVersion
locations[0].raw_type	text
locations[0].license_id
locations[0].is_accepted	False
locations[0].is_published	False
locations[0].raw_source_name
locations[0].landing_page_url	http://arxiv.org/abs/2507.05530
locations[1].id	doi:10.48550/arxiv.2507.05530
locations[1].is_oa	True
locations[1].source.id	https://openalex.org/S4306400194
locations[1].source.issn
locations[1].source.type	repository
locations[1].source.is_oa	True
locations[1].source.issn_l
locations[1].source.is_core	False
locations[1].source.is_in_doaj	False
locations[1].source.display_name	arXiv (Cornell University)
locations[1].source.host_organization	https://openalex.org/I205783295
locations[1].source.host_organization_name	Cornell University
locations[1].source.host_organization_lineage	https://openalex.org/I205783295
locations[1].license	public-domain
locations[1].pdf_url
locations[1].version
locations[1].raw_type	article
locations[1].license_id	https://openalex.org/licenses/public-domain
locations[1].is_accepted	False
locations[1].is_published
locations[1].raw_source_name
locations[1].landing_page_url	https://doi.org/10.48550/arxiv.2507.05530
indexed_in	arxiv, datacite
authorships[0].author.id	https://openalex.org/A5049537298
authorships[0].author.orcid	https://orcid.org/0000-0002-8304-5225
authorships[0].author.display_name	Xiaoping Geng
authorships[0].author_position	first
authorships[0].raw_author_name	Geng, Xi
authorships[0].is_corresponding	False
authorships[1].author.id	https://openalex.org/A5104083508
authorships[1].author.orcid
authorships[1].author.display_name	Cheng Ouyang
authorships[1].author_position	last
authorships[1].raw_author_name	Ouyang, Cheng
authorships[1].is_corresponding	False
has_content.pdf	False
has_content.grobid_xml	False
is_paratext	False
open_access.is_oa	True
open_access.oa_url	https://arxiv.org/pdf/2507.05530
open_access.oa_status	green
open_access.any_repository_has_fulltext	False
created_date	2025-10-10T00:00:00
display_name	Parabolic Anderson Model in Hyperbolic Spaces and Phase Transition
has_fulltext	False
is_retracted	False
updated_date	2025-11-28T10:27:03.454620
primary_topic
cited_by_count	0
locations_count	2
best_oa_location.id	pmh:oai:arXiv.org:2507.05530
best_oa_location.is_oa	True
best_oa_location.source.id	https://openalex.org/S4306400194
best_oa_location.source.issn
best_oa_location.source.type	repository
best_oa_location.source.is_oa	True
best_oa_location.source.issn_l
best_oa_location.source.is_core	False
best_oa_location.source.is_in_doaj	False
best_oa_location.source.display_name	arXiv (Cornell University)
best_oa_location.source.host_organization	https://openalex.org/I205783295
best_oa_location.source.host_organization_name	Cornell University
best_oa_location.source.host_organization_lineage	https://openalex.org/I205783295
best_oa_location.license
best_oa_location.pdf_url	https://arxiv.org/pdf/2507.05530
best_oa_location.version	submittedVersion
best_oa_location.raw_type	text
best_oa_location.license_id
best_oa_location.is_accepted	False
best_oa_location.is_published	False
best_oa_location.raw_source_name
best_oa_location.landing_page_url	http://arxiv.org/abs/2507.05530
primary_location.id	pmh:oai:arXiv.org:2507.05530
primary_location.is_oa	True
primary_location.source.id	https://openalex.org/S4306400194
primary_location.source.issn
primary_location.source.type	repository
primary_location.source.is_oa	True
primary_location.source.issn_l
primary_location.source.is_core	False
primary_location.source.is_in_doaj	False
primary_location.source.display_name	arXiv (Cornell University)
primary_location.source.host_organization	https://openalex.org/I205783295
primary_location.source.host_organization_name	Cornell University
primary_location.source.host_organization_lineage	https://openalex.org/I205783295
primary_location.license
primary_location.pdf_url	https://arxiv.org/pdf/2507.05530
primary_location.version	submittedVersion
primary_location.raw_type	text
primary_location.license_id
primary_location.is_accepted	False
primary_location.is_published	False
primary_location.raw_source_name
primary_location.landing_page_url	http://arxiv.org/abs/2507.05530
publication_date	2025-07-07
publication_year	2025
referenced_works_count	0
abstract_inverted_index.a	1, 64, 104, 115, 123, 128, 156
abstract_inverted_index.In	27
abstract_inverted_index.We	145
abstract_inverted_index.at	103, 127
abstract_inverted_index.be	137
abstract_inverted_index.in	12, 16, 74, 80, 95, 122, 161, 173
abstract_inverted_index.is	10, 36, 44, 63, 150
abstract_inverted_index.it	35
abstract_inverted_index.of	52, 106, 130
abstract_inverted_index.on	114
abstract_inverted_index.to	76, 101
abstract_inverted_index.we	154
abstract_inverted_index.PAM	113
abstract_inverted_index.The	108
abstract_inverted_index.and	14
abstract_inverted_index.for	42, 47, 60, 141, 159
abstract_inverted_index.one	132
abstract_inverted_index.the	19, 39, 49, 53, 68, 83, 92, 112, 138, 142, 163, 170
abstract_inverted_index.$2$,	34
abstract_inverted_index.$2$.	45
abstract_inverted_index.$t$,	131
abstract_inverted_index.$α$	43
abstract_inverted_index.This	87
abstract_inverted_index.fact	93
abstract_inverted_index.from	67, 91, 169
abstract_inverted_index.over	57
abstract_inverted_index.than	33
abstract_inverted_index.that	9, 38, 94, 119, 134, 147
abstract_inverted_index.this	148
abstract_inverted_index.time	13, 75, 81
abstract_inverted_index.when	82
abstract_inverted_index.with	6, 24, 30
abstract_inverted_index.work	110
abstract_inverted_index.$α$.	26
abstract_inverted_index.(PAM)	5
abstract_inverted_index.Given	118
abstract_inverted_index.above	143
abstract_inverted_index.being	71
abstract_inverted_index.grows	55
abstract_inverted_index.model	4
abstract_inverted_index.noise	8
abstract_inverted_index.novel	157
abstract_inverted_index.order	25
abstract_inverted_index.phase	65, 158
abstract_inverted_index.space	125
abstract_inverted_index.speed	105, 129
abstract_inverted_index.tends	100
abstract_inverted_index.there	62
abstract_inverted_index.time,	58
abstract_inverted_index.value	41, 140
abstract_inverted_index.where	18
abstract_inverted_index.which	162
abstract_inverted_index.while	59
abstract_inverted_index.white	11
abstract_inverted_index.would	136
abstract_inverted_index.$α=1$	135
abstract_inverted_index.arises	90
abstract_inverted_index.decays	22
abstract_inverted_index.growth	79
abstract_inverted_index.indeed	151
abstract_inverted_index.moment	51, 70, 165
abstract_inverted_index.motion	99, 121
abstract_inverted_index.second	50, 69, 164
abstract_inverted_index.space,	17, 97
abstract_inverted_index.space.	117, 175
abstract_inverted_index.spaces	29
abstract_inverted_index.bounded	73
abstract_inverted_index.colored	15
abstract_inverted_index.confirm	146
abstract_inverted_index.expects	133
abstract_inverted_index.greater	32
abstract_inverted_index.inverse	84
abstract_inverted_index.present	109
abstract_inverted_index.spatial	20
abstract_inverted_index.travels	126
abstract_inverted_index.uncover	155
abstract_inverted_index.Anderson	3
abstract_inverted_index.Brownian	98, 120
abstract_inverted_index.Consider	0
abstract_inverted_index.Gaussian	7
abstract_inverted_index.behavior	89, 171
abstract_inverted_index.correct.	152
abstract_inverted_index.critical	40, 88, 139
abstract_inverted_index.distinct	168
abstract_inverted_index.explodes	166
abstract_inverted_index.explores	111
abstract_inverted_index.infinity	102
abstract_inverted_index.observed	172
abstract_inverted_index.solution	54
abstract_inverted_index.$α<1$	160
abstract_inverted_index.Euclidean	28, 96, 174
abstract_inverted_index.Parabolic	2
abstract_inverted_index.dimension	31
abstract_inverted_index.intuition	149
abstract_inverted_index.uniformly	72
abstract_inverted_index.$α>2$,	61
abstract_inverted_index.$α<2$,	48
abstract_inverted_index.exhibiting	77
abstract_inverted_index.hyperbolic	116, 124
abstract_inverted_index.increases.	86
abstract_inverted_index.phenomena.	144
abstract_inverted_index.$\sqrt{t}$.	107
abstract_inverted_index.correlation	21
abstract_inverted_index.exponential	78
abstract_inverted_index.temperature	85
abstract_inverted_index.transition,	66
abstract_inverted_index.Furthermore,	153
abstract_inverted_index.polynomially	23
abstract_inverted_index.Specifically,	46
abstract_inverted_index.exponentially	56
abstract_inverted_index.well-understood	37
abstract_inverted_index.sub-exponentially,	167
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	2
citation_normalized_percentile