{"id":199686,"date":"2010-01-04T09:53:03","date_gmt":"2010-01-04T09:53:03","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/events\/northwest-probability-seminar-2010\/"},"modified":"2022-08-31T13:03:20","modified_gmt":"2022-08-31T20:03:20","slug":"northwest-probability-seminar-2010","status":"publish","type":"msr-event","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/event\/northwest-probability-seminar-2010\/","title":{"rendered":"Northwest Probability Seminar 2010"},"content":{"rendered":"\n\n\n\n\n

This is a recap of the 12th Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington (opens in new tab)<\/span><\/a>, the Oregon State University (opens in new tab)<\/span><\/a>, the University of British Columbia (opens in new tab)<\/span><\/a>, the University of Oregon (opens in new tab)<\/span><\/a>, and the Theory Group<\/a> at Microsoft Research<\/a>.  The conference was hosted at Microsoft.<\/p>\n

\"Northwest<\/a><\/p>\n

Supported by Microsoft Research and the Pacific Institute for the Mathematical Sciences<\/b> (opens in new tab)<\/span><\/a> (PIMS).<\/p>\n

The Birnbaum<\/b> (opens in new tab)<\/span><\/a> Lecture in Probability<\/b> was given by Jean-Fran\u00e7ois Le Gall<\/strong> (opens in new tab)<\/span><\/a> (Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).  [Past Birnbaum speakers (opens in new tab)<\/span><\/a>]  The other speakers will be W\u0142odzimierz Bryc (opens in new tab)<\/span><\/a> (U Cincinnati), Gordon Slade (opens in new tab)<\/span><\/a> (U British Columbia), and Allan Sly (opens in new tab)<\/span><\/a> (Microsoft Research), and Edward Waymire (opens in new tab)<\/span><\/a> (Oregon State U).<\/p>\n

The Scientific Committee for the 12th NW Probability Seminar (2010) consisted of Martin Barlow (opens in new tab)<\/span><\/a> (U British Columbia), Chris Burdzy (opens in new tab)<\/span><\/a> (U Washington), Zhen-Qing Chen (opens in new tab)<\/span><\/a> (U Washington), Yevgeniy Kovchegov (opens in new tab)<\/span><\/a> (Oregon State U), David Levin (opens in new tab)<\/span><\/a> (U Oregon), and Yuval Peres<\/a> (Microsoft).<\/p>\n

Schedule & Recordings & Slides<\/h2>\n\n\n\n\n\n\n\n\n\n\n\n\n
 9:45 \u2013 11:00<\/td>\n<\/td>\n<\/td>\nCoffee and muffins<\/b><\/td>\n<\/tr>\n
11:00 \u2013 11:40<\/td>\n \"sly-waymire\"<\/a><\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nAllan Sly<\/b> (opens in new tab)<\/span><\/a> (Microsoft Research)
\nCritical slowdown for Ising model on the two-dimensional lattice<\/td>\n<\/tr>\n
11:55 \u2013 12:35<\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nEdward Waymire<\/b> (opens in new tab)<\/span><\/a> (Oregon State)
\nInterfacial Phenomena and Skew Diffusion<\/td>\n<\/tr>\n
12:35 \u2013 2:00<\/td>\n<\/td>\n<\/td>\nLunch <\/b>(catered)<\/td>\n<\/tr>\n
 1:35 \u2013 2:00<\/td>\n<\/td>\n<\/td>\nOpen problems<\/b> (overlaps with lunch)<\/td>\n<\/tr>\n
 2:05 \u2013 2:55<\/td>\n\"legall-slade\"<\/span><\/a><\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nJean-Fran\u00e7ois Le Gall <\/b> (opens in new tab)<\/span><\/a>(Orsay)
\nThe continuous limit of large random planar maps<\/td>\n<\/tr>\n
 3:05 \u2013 3:45<\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nGordon Slade<\/b> (opens in new tab)<\/span><\/a> (U British Columbia)
\nA renormalisation group analysis of the
\n4-dimensional continuous-time weakly self-avoiding walk<\/td>\n<\/tr>\n
 3:45 \u2013 4:20<\/td>\n<\/td>\n<\/td>\nTea and snacks<\/b><\/td>\n<\/tr>\n
 4:20 \u2013 5:00<\/td>\n \"bryc\"<\/a><\/td>\n
\n\"pdf\"<\/span><\/a><\/span><\/td>\n		W\u0142odzimierz Bryc<\/b> (opens in new tab)<\/span><\/a> (U Cincinnati)
\nMartingales from pairs of randomized Poisson, Gamma,
\nnegative binomial and hyperbolic secant processes<\/td>\n<\/tr>\n
 5:45 \u2013<\/td>\n<\/td>\n<\/td>\nDinner<\/b> (catered)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n

Critical slowdown for Ising model on the two-dimensional lattice<\/h3>\n

Allan Sly<\/b> (opens in new tab)<\/span><\/a> (Microsoft Research)<\/p>\n

Abstract:<\/em> Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $Z_2$ everywhere except at criticality. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of works verified this picture on $Z_2$ except at $beta=beta_c$ where the behavior remained unknown. In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $Z_2$. Namely, we show that on a finite box with arbitrary boundary conditions, the inverse-gap at $beta=beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.<\/p>\n

Joint work with Eyal Lubetzky.<\/p>\n

Interfacial Phenomena and Skew Diffusion<\/h3>\n

Edward Waymire<\/b> (opens in new tab)<\/span><\/a> (Oregon State)<\/p>\n

Abstract:<\/em> Skew diffusion refers to stochastic processes whose infinitesimal generators are second order advection-dispersion elliptic operators having piecewise constant coefficients. Such processes arise naturally in connection with macroscopic mass balance and flux laws in highly heterogeneous environments. We shall discuss some recent results pertaining to interfacial effects in terms of martingale properties, local time and first passage time properties.<\/p>\n

This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood.<\/p>\n

The continuous limit of large random planar maps<\/h3>\n

Jean-Fran\u00e7ois Le Gall <\/b> (opens in new tab)<\/span><\/a>(Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).<\/p>\n

Abstract:<\/em> Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they also have significant geometrical applications. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n^{-1\/4}. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit, which is called the Brownian map, can be written as a quotient space of Aldous\u2019 Continuum Random Tree (the CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. We discuss various properties of the Brownian map.<\/p>\n

A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk<\/h3>\n

Gordon Slade<\/b> (opens in new tab)<\/span><\/a> (U British Columbia)<\/p>\n

Abstract:<\/em> We discuss recent joint work with David Brydges which proves |x|^{-2} decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z^4. The walk two-point function is identified as the two-point function of a supersymmetric field theory with quartic self-interaction, and the field theory is then analysed using renormalisation group methods.<\/p>\n

Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes<\/h3>\n

W\u0142odzimierz Bryc<\/b> (opens in new tab)<\/span><\/a> (U Cincinnati)<\/p>\n

Abstract:<\/em> Consider a pair of independent Poisson processes, or a pair of Negative Binomial processes, or Gamma, or hyperbolic secant processes with a shared randomly selected parameter. Under appropriate randomization, one can deterministically re-parametrize the time and scale for both processes so that the first process runs on time interval $(0,1)$, the second process runs on time interval $(1,infty)$, and the two processes seamlessly join into one Markov martingale on $(0,infty)$. In fact, a property stronger than martingale holds: we stitch together two processes into a single quadratic harness on $(0,infty)$.<\/p>\n

This talk is based on joint work in progress with J. Wesolowski.<\/p>\n\n\n","protected":false},"excerpt":{"rendered":"

This is a recap of the 12th Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington (opens in new tab), the Oregon State University (opens in new tab), the University of British Columbia (opens in new tab), the University of Oregon (opens in new tab), and the Theory Group at Microsoft Research.  The conference was […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_startdate":"2010-10-16","msr_enddate":"2010-10-16","msr_location":"Redmond, WA, U.S.","msr_expirationdate":"","msr_event_recording_link":"","msr_event_link":"","msr_event_link_redirect":false,"msr_event_time":"","msr_hide_region":false,"msr_private_event":true,"msr_hide_image_in_river":0,"footnotes":""},"research-area":[13561],"msr-region":[197900],"msr-event-type":[197941],"msr-video-type":[],"msr-locale":[268875],"msr-program-audience":[],"msr-post-option":[],"msr-impact-theme":[],"class_list":["post-199686","msr-event","type-msr-event","status-publish","hentry","msr-research-area-algorithms","msr-region-north-america","msr-event-type-conferences","msr-locale-en_us"],"msr_about":"\n\n\n\n\n

This is a recap of the 12th Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington<\/a>, the Oregon State University<\/a>, the University of British Columbia<\/a>, the University of Oregon<\/a>, and the Theory Group<\/a> at Microsoft Research<\/a>.  The conference was hosted at Microsoft.<\/p>\n

\"Northwest<\/a><\/p>\n

Supported by Microsoft Research and the Pacific Institute for the Mathematical Sciences<\/b><\/a> (PIMS).<\/p>\n

The Birnbaum<\/b><\/a> Lecture in Probability<\/b> was given by Jean-Fran\u00e7ois Le Gall<\/strong><\/a> (Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).  [Past Birnbaum speakers<\/a>]  The other speakers will be W\u0142odzimierz Bryc<\/a> (U Cincinnati), Gordon Slade<\/a> (U British Columbia), and Allan Sly<\/a> (Microsoft Research), and Edward Waymire<\/a> (Oregon State U).<\/p>\n

The Scientific Committee for the 12th NW Probability Seminar (2010) consisted of Martin Barlow<\/a> (U British Columbia), Chris Burdzy<\/a> (U Washington), Zhen-Qing Chen<\/a> (U Washington), Yevgeniy Kovchegov<\/a> (Oregon State U), David Levin<\/a> (U Oregon), and Yuval Peres<\/a> (Microsoft).<\/p>\n

Schedule & Recordings & Slides<\/h2>\n\n\n\n\n\n\n\n\n\n\n\n\n
 9:45 \u2013 11:00<\/td>\n<\/td>\n<\/td>\nCoffee and muffins<\/b><\/td>\n<\/tr>\n
11:00 \u2013 11:40<\/td>\n \"sly-waymire\"<\/a><\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nAllan Sly<\/b><\/a> (Microsoft Research)
\nCritical slowdown for Ising model on the two-dimensional lattice<\/td>\n<\/tr>\n
11:55 \u2013 12:35<\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nEdward Waymire<\/b><\/a> (Oregon State)
\nInterfacial Phenomena and Skew Diffusion<\/td>\n<\/tr>\n
12:35 \u2013 2:00<\/td>\n<\/td>\n<\/td>\nLunch <\/b>(catered)<\/td>\n<\/tr>\n
 1:35 \u2013 2:00<\/td>\n<\/td>\n<\/td>\nOpen problems<\/b> (overlaps with lunch)<\/td>\n<\/tr>\n
 2:05 \u2013 2:55<\/td>\n\"legall-slade\"<\/span><\/a><\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nJean-Fran\u00e7ois Le Gall <\/b><\/a>(Orsay)
\nThe continuous limit of large random planar maps<\/td>\n<\/tr>\n
 3:05 \u2013 3:45<\/td>\n\"pdf\"<\/span><\/span><\/a><\/td>\nGordon Slade<\/b><\/a> (U British Columbia)
\nA renormalisation group analysis of the
\n4-dimensional continuous-time weakly self-avoiding walk<\/td>\n<\/tr>\n
 3:45 \u2013 4:20<\/td>\n<\/td>\n<\/td>\nTea and snacks<\/b><\/td>\n<\/tr>\n
 4:20 \u2013 5:00<\/td>\n \"bryc\"<\/a><\/td>\n
\n\"pdf\"<\/span><\/a><\/span><\/td>\n		W\u0142odzimierz Bryc<\/b><\/a> (U Cincinnati)
\nMartingales from pairs of randomized Poisson, Gamma,
\nnegative binomial and hyperbolic secant processes<\/td>\n<\/tr>\n
 5:45 \u2013<\/td>\n<\/td>\n<\/td>\nDinner<\/b> (catered)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n

Critical slowdown for Ising model on the two-dimensional lattice<\/h3>\n

Allan Sly<\/b><\/a> (Microsoft Research)<\/p>\n

Abstract:<\/em> Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $Z_2$ everywhere except at criticality. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of works verified this picture on $Z_2$ except at $beta=beta_c$ where the behavior remained unknown. In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $Z_2$. Namely, we show that on a finite box with arbitrary boundary conditions, the inverse-gap at $beta=beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.<\/p>\n

Joint work with Eyal Lubetzky.<\/p>\n

Interfacial Phenomena and Skew Diffusion<\/h3>\n

Edward Waymire<\/b><\/a> (Oregon State)<\/p>\n

Abstract:<\/em> Skew diffusion refers to stochastic processes whose infinitesimal generators are second order advection-dispersion elliptic operators having piecewise constant coefficients. Such processes arise naturally in connection with macroscopic mass balance and flux laws in highly heterogeneous environments. We shall discuss some recent results pertaining to interfacial effects in terms of martingale properties, local time and first passage time properties.<\/p>\n

This is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood.<\/p>\n

The continuous limit of large random planar maps<\/h3>\n

Jean-Fran\u00e7ois Le Gall <\/b><\/a>(Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).<\/p>\n

Abstract:<\/em> Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they also have significant geometrical applications. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n^{-1\/4}. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit, which is called the Brownian map, can be written as a quotient space of Aldous\u2019 Continuum Random Tree (the CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. We discuss various properties of the Brownian map.<\/p>\n

A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk<\/h3>\n

Gordon Slade<\/b><\/a> (U British Columbia)<\/p>\n

Abstract:<\/em> We discuss recent joint work with David Brydges which proves |x|^{-2} decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z^4. The walk two-point function is identified as the two-point function of a supersymmetric field theory with quartic self-interaction, and the field theory is then analysed using renormalisation group methods.<\/p>\n

Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes<\/h3>\n

W\u0142odzimierz Bryc<\/b><\/a> (U Cincinnati)<\/p>\n

Abstract:<\/em> Consider a pair of independent Poisson processes, or a pair of Negative Binomial processes, or Gamma, or hyperbolic secant processes with a shared randomly selected parameter. Under appropriate randomization, one can deterministically re-parametrize the time and scale for both processes so that the first process runs on time interval $(0,1)$, the second process runs on time interval $(1,infty)$, and the two processes seamlessly join into one Markov martingale on $(0,infty)$. In fact, a property stronger than martingale holds: we stitch together two processes into a single quadratic harness on $(0,infty)$.<\/p>\n

This talk is based on joint work in progress with J. Wesolowski.<\/p>\n\n\n","tab-content":[{"id":0,"name":"Summary","content":"This is a recap of the\u00a012th\u00a0Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington<\/a>, the Oregon State University<\/a>, the University of British Columbia<\/a>, the University of Oregon<\/a>, and the Theory Group<\/a> at Microsoft Research<\/a>.\u00a0 The conference\u00a0was hosted at Microsoft.\r\n\r\n\"Northwest<\/a>\r\n\r\nSupported by Microsoft Research and the Pacific Institute for the Mathematical Sciences<\/b><\/a> (PIMS).\r\n\r\nThe Birnbaum<\/b><\/a> Lecture in Probability<\/b> was given by Jean-Fran\u00e7ois Le Gall<\/strong><\/a> (Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).\u00a0 [Past Birnbaum speakers<\/a>]\u00a0 The other speakers will\u00a0be W\u0142odzimierz Bryc<\/a>\u00a0(U Cincinnati), Gordon Slade<\/a>\u00a0(U British Columbia), and Allan Sly<\/a>\u00a0(Microsoft Research), and Edward Waymire<\/a> (Oregon State U).\r\n\r\nThe Scientific Committee for the\u00a012th NW Probability Seminar (2010) consisted of Martin Barlow<\/a> (U British Columbia), Chris Burdzy<\/a> (U Washington), Zhen-Qing Chen<\/a> (U Washington), Yevgeniy Kovchegov<\/a> (Oregon State U), David Levin<\/a> (U Oregon), and Yuval Peres<\/a> (Microsoft).\r\n

Schedule & Recordings & Slides<\/h2>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a09:45 - 11:00<\/td>\r\n<\/td>\r\n<\/td>\r\nCoffee and muffins<\/b><\/td>\r\n<\/tr>\r\n
11:00 - 11:40<\/td>\r\n\u00a0\"sly-waymire\"<\/a><\/td>\r\n\"pdf\"<\/span><\/span><\/a><\/td>\r\nAllan Sly<\/b><\/a> (Microsoft Research)\r\nCritical slowdown for Ising model on the two-dimensional lattice<\/td>\r\n<\/tr>\r\n
11:55 - 12:35<\/td>\r\n\"pdf\"<\/span><\/span><\/a><\/td>\r\nEdward Waymire<\/b><\/a> (Oregon State)\r\nInterfacial Phenomena and Skew Diffusion<\/td>\r\n<\/tr>\r\n
12:35 - 2:00<\/td>\r\n<\/td>\r\n<\/td>\r\nLunch <\/b>(catered)<\/td>\r\n<\/tr>\r\n
\u00a01:35 - 2:00<\/td>\r\n<\/td>\r\n<\/td>\r\nOpen problems<\/b> (overlaps with lunch)<\/td>\r\n<\/tr>\r\n
\u00a02:05 - 2:55<\/td>\r\n\"legall-slade\"<\/span><\/a><\/td>\r\n\"pdf\"<\/span><\/span><\/a><\/td>\r\nJean-Fran\u00e7ois Le Gall <\/b><\/a>(Orsay)\r\nThe continuous limit of large random planar maps<\/td>\r\n<\/tr>\r\n
\u00a03:05 - 3:45<\/td>\r\n\"pdf\"<\/span><\/span><\/a><\/td>\r\nGordon Slade<\/b><\/a> (U British Columbia)\r\nA renormalisation group analysis of the\r\n4-dimensional continuous-time weakly self-avoiding walk<\/td>\r\n<\/tr>\r\n
\u00a03:45 - 4:20<\/td>\r\n<\/td>\r\n<\/td>\r\nTea and snacks<\/b><\/td>\r\n<\/tr>\r\n
\u00a04:20 - 5:00<\/td>\r\n\u00a0\"bryc\"<\/a><\/td>\r\n\r\n\"pdf\"<\/span><\/a><\/span><\/td>\r\nW\u0142odzimierz Bryc<\/b><\/a> (U Cincinnati)\r\nMartingales from pairs of randomized Poisson, Gamma,\r\nnegative binomial and hyperbolic secant processes<\/td>\r\n<\/tr>\r\n
\u00a05:45 -<\/td>\r\n<\/td>\r\n<\/td>\r\nDinner<\/b>\u00a0(catered)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>"},{"id":1,"name":"Talk Abstracts","content":"

Critical slowdown for Ising model on the two-dimensional lattice<\/h3>\r\nAllan Sly<\/b><\/a> (Microsoft Research)\r\n\r\nAbstract:<\/em> Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $\\Z_2$ everywhere except at criticality. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $\\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of works verified this picture on $\\Z_2$ except at $\\beta=\\beta_c$ where the behavior remained unknown. In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $\\Z_2$. Namely, we show that on a finite box with arbitrary boundary conditions, the inverse-gap at $\\beta=\\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.\r\n\r\nJoint work with Eyal Lubetzky.\r\n

Interfacial Phenomena and Skew Diffusion<\/h3>\r\nEdward Waymire<\/b><\/a> (Oregon State)\r\n\r\nAbstract:<\/em> Skew diffusion refers to stochastic processes whose infinitesimal generators are second order advection-dispersion elliptic operators having piecewise constant coefficients. Such processes arise naturally in connection with macroscopic mass balance and flux laws in highly heterogeneous environments. We shall discuss some recent results pertaining to interfacial effects in terms of martingale properties, local time and first passage time properties.\r\n\r\nThis is based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, and Brian Wood.\r\n

The continuous limit of large random planar maps<\/h3>\r\nJean-Fran\u00e7ois Le Gall <\/b><\/a>(Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).\r\n\r\nAbstract:<\/em> Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they also have significant geometrical applications. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n^{-1\/4}. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit, which is called the Brownian map, can be written as a quotient space of Aldous' Continuum Random Tree (the CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. We discuss various properties of the Brownian map.\r\n

A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk<\/h3>\r\nGordon Slade<\/b><\/a> (U British Columbia)\r\n\r\nAbstract:<\/em> We discuss recent joint work with David Brydges which proves |x|^{-2} decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z^4. The walk two-point function is identified as the two-point function of a supersymmetric field theory with quartic self-interaction, and the field theory is then analysed using renormalisation group methods.\r\n

Martingales from pairs of randomized Poisson, Gamma, negative binomial and hyperbolic secant processes<\/h3>\r\nW\u0142odzimierz Bryc<\/b><\/a> (U Cincinnati)\r\n\r\nAbstract:<\/em> Consider a pair of independent Poisson processes, or a pair of Negative Binomial processes, or Gamma, or hyperbolic secant processes with a shared randomly selected parameter. Under appropriate randomization, one can deterministically re-parametrize the time and scale for both processes so that the first process runs on time interval $(0,1)$, the second process runs on time interval $(1,\\infty)$, and the two processes seamlessly join into one Markov martingale on $(0,\\infty)$. In fact, a property stronger than martingale holds: we stitch together two processes into a single quadratic harness on $(0,\\infty)$.\r\n\r\nThis talk is based on joint work in progress with J. Wesolowski."}],"msr_startdate":"2010-10-16","msr_enddate":"2010-10-16","msr_event_time":"","msr_location":"Redmond, WA, U.S.","msr_event_link":"","msr_event_recording_link":"","msr_startdate_formatted":"October 16, 2010","msr_register_text":"Watch now","msr_cta_link":"","msr_cta_text":"","msr_cta_bi_name":"","featured_image_thumbnail":null,"event_excerpt":"This is a recap of the\u00a012th\u00a0Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington, the Oregon State University, the University of British Columbia, the University of Oregon, and the Theory Group at Microsoft Research.\u00a0 The conference\u00a0was hosted at Microsoft. Supported by Microsoft Research and the Pacific Institute for the Mathematical Sciences (PIMS). The Birnbaum Lecture in Probability was given by Jean-Fran\u00e7ois Le Gall (Universit\u00e9 Paris-Sud, Orsay and Institut Universitaire de France).\u00a0 [Past…","msr_research_lab":[199565],"related-researchers":[],"msr_impact_theme":[],"related-academic-programs":[],"related-groups":[],"related-projects":[],"related-opportunities":[],"related-publications":[],"related-videos":[],"related-posts":[],"_links":{"self":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-event\/199686","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-event"}],"about":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-event"}],"version-history":[{"count":2,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-event\/199686\/revisions"}],"predecessor-version":[{"id":874284,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-event\/199686\/revisions\/874284"}],"wp:attachment":[{"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/media?parent=199686"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=199686"},{"taxonomy":"msr-region","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-region?post=199686"},{"taxonomy":"msr-event-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-event-type?post=199686"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=199686"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=199686"},{"taxonomy":"msr-program-audience","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-program-audience?post=199686"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=199686"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=199686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}