The k-tacnode process
The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any k ∈ Z , we show...
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| Veröffentlicht in: | Probability theory and related fields 2019-10, Vol.175 (1-2), p.341-395 |
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| container_title | Probability theory and related fields |
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| creator | Buckingham, Robert Liechty, Karl |
| description | The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any
k
∈
Z
, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which
k
particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight. |
| doi_str_mv | 10.1007/s00440-018-0885-2 |
| format | Article |
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k
∈
Z
, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which
k
particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.</description><identifier>ISSN: 0178-8051</identifier><identifier>EISSN: 1432-2064</identifier><identifier>DOI: 10.1007/s00440-018-0885-2</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Asymptotic methods ; Brownian motion ; Double cusps ; Drift ; Economics ; Finance ; Insurance ; Management ; Mathematical analysis ; Mathematical and Computational Biology ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Operations Research/Decision Theory ; Polynomials ; Probability ; Probability Theory and Stochastic Processes ; Quantitative Finance ; Statistics for Business ; Theoretical ; Tiling ; Weight</subject><ispartof>Probability theory and related fields, 2019-10, Vol.175 (1-2), p.341-395</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2018</rights><rights>Probability Theory and Related Fields is a copyright of Springer, (2018). All Rights Reserved.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-1e8a6e014e87212792fa7f1c1c044eb7cf4f7c0ad8675895a54c339a305c7b353</citedby><cites>FETCH-LOGICAL-c316t-1e8a6e014e87212792fa7f1c1c044eb7cf4f7c0ad8675895a54c339a305c7b353</cites><orcidid>0000-0001-6399-5920</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00440-018-0885-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00440-018-0885-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27915,27916,41479,42548,51310</link.rule.ids></links><search><creatorcontrib>Buckingham, Robert</creatorcontrib><creatorcontrib>Liechty, Karl</creatorcontrib><title>The k-tacnode process</title><title>Probability theory and related fields</title><addtitle>Probab. Theory Relat. Fields</addtitle><description>The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any
k
∈
Z
, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which
k
particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.</description><subject>Asymptotic methods</subject><subject>Brownian motion</subject><subject>Double cusps</subject><subject>Drift</subject><subject>Economics</subject><subject>Finance</subject><subject>Insurance</subject><subject>Management</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Biology</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>Polynomials</subject><subject>Probability</subject><subject>Probability Theory and Stochastic Processes</subject><subject>Quantitative Finance</subject><subject>Statistics for Business</subject><subject>Theoretical</subject><subject>Tiling</subject><subject>Weight</subject><issn>0178-8051</issn><issn>1432-2064</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNp1jz1PAzEQRC0EEiFQUtBFojbsru2zr0QRX1IkmlBbjrMGAtwF-1Lw73F0SFRU27yZ2SfEBcIVAtjrAqA1SEAnwTkj6UBMUCuSBI0-FBNA66QDg8fipJQNAJDSNBHny1eevcshxK5f82yb-8ilnIqjFD4Kn_3eqXi-u13OH-Ti6f5xfrOQUWEzSGQXGgbU7Cwh2ZZSsAkjxvoMr2xMOtkIYe0aa1xrgtFRqTYoMNGulFFTcTn21t2vHZfBb_pd7uqkJ9StMa1TqlI4UjH3pWROfpvfPkP-9gh-b-9He1_t_d7eU83QmCmV7V44_zX_H_oBvq1Zew</recordid><startdate>20191001</startdate><enddate>20191001</enddate><creator>Buckingham, Robert</creator><creator>Liechty, Karl</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>H8D</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2O</scope><scope>M2P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0001-6399-5920</orcidid></search><sort><creationdate>20191001</creationdate><title>The k-tacnode process</title><author>Buckingham, Robert ; Liechty, Karl</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-1e8a6e014e87212792fa7f1c1c044eb7cf4f7c0ad8675895a54c339a305c7b353</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Asymptotic methods</topic><topic>Brownian motion</topic><topic>Double cusps</topic><topic>Drift</topic><topic>Economics</topic><topic>Finance</topic><topic>Insurance</topic><topic>Management</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Biology</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>Polynomials</topic><topic>Probability</topic><topic>Probability Theory and Stochastic Processes</topic><topic>Quantitative Finance</topic><topic>Statistics for Business</topic><topic>Theoretical</topic><topic>Tiling</topic><topic>Weight</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Buckingham, Robert</creatorcontrib><creatorcontrib>Liechty, Karl</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central</collection><collection>Advanced Technologies & Aerospace Database (1962 - current)</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>ProQuest Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>Aerospace Database</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ABI/INFORM Global</collection><collection>Computing Database</collection><collection>ProQuest research library</collection><collection>Science Database (ProQuest)</collection><collection>ProQuest Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>ProQuest Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Probability theory and related fields</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Buckingham, Robert</au><au>Liechty, Karl</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The k-tacnode process</atitle><jtitle>Probability theory and related fields</jtitle><stitle>Probab. Theory Relat. Fields</stitle><date>2019-10-01</date><risdate>2019</risdate><volume>175</volume><issue>1-2</issue><spage>341</spage><epage>395</epage><pages>341-395</pages><issn>0178-8051</issn><eissn>1432-2064</eissn><abstract>The tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems. For Dyson Brownian bridges, the tacnode process describes the grazing collision of two packets of walkers. We consider such a Dyson sea on the unit circle with drift. For any
k
∈
Z
, we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which
k
particles are expected to wrap around the circle. We derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation. The method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00440-018-0885-2</doi><tpages>55</tpages><orcidid>https://orcid.org/0000-0001-6399-5920</orcidid></addata></record> |
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| ispartof | Probability theory and related fields, 2019-10, Vol.175 (1-2), p.341-395 |
| issn | 0178-8051 1432-2064 |
| language | eng |
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| source | Business Source Complete; SpringerLink Journals - AutoHoldings |
| subjects | Asymptotic methods Brownian motion Double cusps Drift Economics Finance Insurance Management Mathematical analysis Mathematical and Computational Biology Mathematical and Computational Physics Mathematics Mathematics and Statistics Operations Research/Decision Theory Polynomials Probability Probability Theory and Stochastic Processes Quantitative Finance Statistics for Business Theoretical Tiling Weight |
| title | The k-tacnode process |
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