Difference between revisions of "Probability Seminar"

From UW-Math Wiki
Jump to navigation Jump to search
 
(46 intermediate revisions by 6 users not shown)
Line 1: Line 1:
 
__NOTOC__
 
__NOTOC__
 +
[[Probability | Back to Probability Group]]
  
= Fall 2021 =
+
= Fall 2022 =
  
 
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>  
 
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>  
  
We usually end for questions at 3:20 PM.
+
We usually end for questions at 3:20 PM.
  
 
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
 
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
  
 
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
 
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
 +
 +
 +
== September 22, 2022, in person: [https://sites.google.com/site/pierreyvesgl/home Pierre Yves Gaudreau Lamarre] (University of Chicago)    ==
 +
 +
'''Moments of the Parabolic Anderson Model with Asymptotically Singular Noise'''
 
   
 
   
== September 16, 2021, in person: [https://hanbaeklyu.com/ Hanbaek Lyu] (UW-Madison)   ==  
+
The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.
 +
 +
One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.
 +
 +
In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.
 +
 +
This talk is based on a joint work with Promit Ghosal and Yuchen Liao.
 +
 
 +
== September 29, 2022, in person: Christian Gorski (Northwestern University)   ==
  
 +
'''Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees'''
  
'''Scaling limit of soliton statistics of a multicolor box-ball system'''
+
I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance.
 +
If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.
  
The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.
+
== October 6, 2022, in person: [https://danielslonim.github.io/ Daniel Slonim] (University of Virginia)  ==
  
== September 23, 2021, no seminar  ==
+
'''Random Walks in (Dirichlet) Random Environments with Jumps on Z'''
  
 +
We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.
  
== September 30, 2021, in person: [https://mrusskikh.mit.edu/home Marianna Russskikh] (MIT)   ==
+
== October 13, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.univ-evry.fr/pages_perso/loukianova/ Dasha Loukianova] (Université d'Évry Val d'Essonne)   ==
  
'''Lozenge tilings and the Gaussian free field on a cylinder'''
+
== October 20, 2022, '''4pm, VV911''', in person: [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
 +
''Note the unusual time and room!''
  
We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.
+
'''An introduction to counts-of-counts data'''
  
== October 7, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://n.ethz.ch/~bdembin/home.html Barbara Dembin] (ETH Zurich)    ==
+
Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size
  
 +
''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>''  containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.
  
== October 14, 2021, in person: [https://sites.google.com/wisc.edu/evan-sorensen Evan Sorensen] (UW-Madison)  ==
+
In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.
  
  
== October 21, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [http://stat.columbia.edu/~sumitm/ Sumit Mukherjee] (Columbia)    ==
+
''References''
  
 +
[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943
  
 +
[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
  
== October 28, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~wkchen/ Wei-Kuo Chen] (Minnesota)    ==
+
[3] Ewens WJ. Theoret Popul Biol, 3, 1972
  
 +
[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)
  
== November 4, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://mathematics.stanford.edu/people/mackenzie-simper Mackenzie Simper] (Stanford)  ==
 
  
  
 +
== October 27, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~arnab/ Arnab Sen] (University of Minnesota, Twin Cities) ==
  
== November 11, 2021, in person: [http://web.lfzhang.com/ Lingfu Zhang] (Princeton)    ==
 
  
== November 18, 2021, in person [http://www.ilt.kharkov.ua/bvi/structure/depart_e/d24/mariya_shcherbina-cv.htm Mariya Shcherbina] (Kharkov) ==
+
== November 3, 2022, in person: [https://www.ias.edu/scholars/sky-yang-cao Sky Cao] (Institute for Advanced Study)   ==  
  
== November 25, 2021, no seminar  ==
 
  
 +
== November 10, 2022, in person: TBD  ==
  
== December 2, 2021, in person: [http://math.uchicago.edu/~xuanw/ Xuan Wu] (Chicago)  ==
 
  
 +
== November 17, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/site/leandroprpimentel/ Leandro Pimentel] (Federal University of Rio de Janeiro)  ==
  
== December 9, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.dur.ac.uk/users/sunil.chhita/ Sunil Chhita] (Durham)  ==
 
  
'''GOE Fluctuations for the maximum of the top path in ASMs'''
+
== December 1, in person: [https://cims.nyu.edu/~ajd594/ Alex Dunlap] (Courant Institute)  ==
  
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling.  This talk is based on joint work with Arvind Ayyer and Kurt Johansson.
 
  
 +
== December 8, 2022, in person: [https://sites.northwestern.edu/juliagaudio/ Julia Gaudio] (Northwestern University)  ==
  
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 13:53, 30 September 2022

Back to Probability Group

Fall 2022

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

ZOOM LINK. Valid only for online seminars.

If you would like to sign up for the email list to receive seminar announcements then please join our group.


September 22, 2022, in person: Pierre Yves Gaudreau Lamarre (University of Chicago)

Moments of the Parabolic Anderson Model with Asymptotically Singular Noise

The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.

One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.

In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.

This talk is based on a joint work with Promit Ghosal and Yuchen Liao.

September 29, 2022, in person: Christian Gorski (Northwestern University)

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance. If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.

October 6, 2022, in person: Daniel Slonim (University of Virginia)

Random Walks in (Dirichlet) Random Environments with Jumps on Z

We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.

October 13, 2022, ZOOM: Dasha Loukianova (Université d'Évry Val d'Essonne)

October 20, 2022, 4pm, VV911, in person: Simon Tavaré (Columbia University)

Note the unusual time and room!

An introduction to counts-of-counts data

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers C1, C2, … of species represented once, twice, … in a sample of size

N = C1 + 2 C2 + 3 C3 + ….  containing S = C1 + C2 + species; the vector C = (C1, C2, …) gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

In this talk I will outline some of the stochastic models used to model the distribution of C, and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of S in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.


References

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. Logarithmic Combinatorial Structures, EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)


October 27, 2022, ZOOM: Arnab Sen (University of Minnesota, Twin Cities)

November 3, 2022, in person: Sky Cao (Institute for Advanced Study)

November 10, 2022, in person: TBD

November 17, 2022, ZOOM: Leandro Pimentel (Federal University of Rio de Janeiro)

December 1, in person: Alex Dunlap (Courant Institute)

December 8, 2022, in person: Julia Gaudio (Northwestern University)

Past Seminars