Probability Seminar: Difference between revisions
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'''Likelihood landscape on a known phylogeny''' | '''Likelihood landscape on a known phylogeny''' | ||
Abstract: Over time, ancestral populations evolve to become separate species. We can represent this history as a tree with edge lengths where the leaves are the modern-day species. If we know the precise topology of the tree (i.e. the precise evolutionary relationship between all the species), then we can imagine traits (their presence or absence) being passed down according to a symmetric 2-state continuous-time Markov chain. The branch length becomes the probability a parent species has a trait while the child species does not. This length is unknown, but researchers have observed they can get pretty good estimates using maximum likelihood estimation and only the leaf data despite the fact that the number of critical points for the log-likelihood grows exponentially fast in the size of the tree. In this talk, I will discuss why this MLE approach works by showing that the population log-likelihood is strictly concave and smooth in a neighborhood around the true branch length parameters and the size. | |||
This talk is based on joint work with Hanbaek Lyu, Sebastien Roch and Allan Sly. | |||
== November | == November 7, 2024: Zoe Huang (UNC Chapel Hill) == | ||
'''Cutoff for Cayley graphs of nilpotent groups''' | |||
== | Abstract: Abstract: We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly bounded. For some $k=k(n)$ such that $1\ll\log k \ll \log |G|$, we pick a random set of generators $S=S(n)$ by sampling $k$ elements $Z_1,\ldots,Z_k$ from $G$ uniformly at random with replacement, and set $S:=\{Z_j^{\pm 1}:1 \le j\le k \}$. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant $c>0$, depending only on the rank and the nilpotency class of $G$, such that for all symmetric sets of generators $S$ of size at most $ \frac{c\log |G|}{\log \log |G|}$, the spectral gap and the $\varepsilon$-mixing time of the simple random walk $X=(X_t)_{t\geq 0}$ on Cay$(G,S)$ are asymptotically the same as those of the projection of $X$ to the abelianization of $G$, given by $[G,G]X_t$. In particular, $X$ exhibits cutoff if and only if its projection does. Based on joint work with Jonathan Hermon. | ||
== | == November 14, 2024: Nabarun Deb (University of Chicago) == | ||
Mean-Field fluctuations in Ising models and posterior prediction intervals in low signal-to-noise ratio regimes | |||
Ising models have become central in probability, statistics, and machine learning. They naturally appear in the posterior distribution of regression coefficients under the linear model $Y = X\beta + \epsilon$, where $\epsilon \sim N(0, \sigma^2 I_n)$. This talk explores fluctuations of specific linear statistics under the Ising model, with a focus on applications in Bayesian linear regression. | |||
In the first part, we examine Ising models on "dense regular" graphs and characterize the limiting distribution of average magnetization across various temperature and magnetization regimes, extending previous results beyond the Curie-Weiss (complete graph) case. In the second part, we analyze posterior prediction intervals for linear statistics in low signal-to-noise ratio (SNR) scenarios, also known as the contiguity regime. Here, unlike standard Bernstein-von Mises results, the limiting distributions are highly sensitive to the choice of prior. We illustrate this dependency by presenting limiting laws under both correctly specified and misspecified priors. | |||
This talk is based on joint work with Sumit Mukherjee and Seunghyun Li. | |||
= | == November 21, 2024: Reza Gheissari (Northwestern) == | ||
'''Wetting and pre-wetting in (2+1)D solid-on-solid interfaces''' | |||
The (d+1)D-solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d\ge 2$, with zero-boundary conditions, at low temperatures the surface is localized about height $0$, but when constrained to take only non-negative values entropic repulsion pushes it to take typical heights of $O(\log n)$. I will describe the mechanism of entropic repulsion, and present results on how the picture changes when one introduces a competing force trying to keep the interface localized (either an external field or a reward for points where the height is exactly zero). Along the way, I will outline rich predictions for the shapes of level curves, and for metastability phenomena in the Glauber dynamics. Based on joint work with Eyal Lubetzky and Joseph Chen. | |||
== | == November 28, 2024: Thanksgiving == | ||
No seminar | |||
== December 5, 2024: Erik Bates (NC State) == | |||
== | |||
'''Parisi formulas in multi-species and vector spin glass models''' | |||
''' | |||
The expression "Parisi formula" refers to a variational formula postulated by Parisi in 1980 to give the limiting free energy of the Sherrington--Kirkpatrick (SK) spin glass. The SK model was originally conceived as a mean-field description for disordered magnetism, and has since become a mathematical prototype for frustrated disordered systems and high-complexity functions. In recent years, there has been an effort to extend the Parisi framework to various generalizations of the SK model, raising new physical questions met with fresh mathematical challenges. In this talk, I will share some developments in this evolving story. Based on joint works with Leila Sloman and Youngtak Sohn. | |||
Latest revision as of 20:39, 22 November 2024
- When: Thursdays at 2:30 pm
- Where: 901 Van Vleck Hall
- Organizers: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
- To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
- To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu
Fall 2024
Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom
We usually end for questions at 3:20 PM.
September 5, 2024:
No seminar
September 12, 2024: Hongchang Ji (UW-Madison)
Spectral edge of non-Hermitian random matrices
We report recent progress on spectra of so-called deformed i.i.d. matrices. They are square non-Hermitian random matrices of the form $A+X$ where $X$ has centered i.i.d. entries and $A$ is a deterministic bias, and $A$ and $X$ are on the same scale so that their contributions to the spectrum of $A+X$ are comparable. Under this setting, we present two recent results concerning universal patterns arising in eigenvalue statistics of $A+X$ around its boundary, on macroscopic and microscopic scales. The first result shows that the macroscopic eigenvalue density of $A+X$ typically has a jump discontinuity around the boundary of its support, which is a distinctive feature of $X$ by the \emph{circular law}. The second result is edge universality for deformed non-Hermitian matrices; it shows that the local eigenvalue statistics of $A+X$ around a typical (jump) boundary point is universal, i.e., matches with those of a Ginibre matrix $X$ with i.i.d. standard Gaussian entries.
Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.
September 19, 2024: Miklos Racz (Northwestern)
The largest common subtree of uniform attachment trees
Consider two independent uniform attachment trees with n nodes each -- how large is their largest common subtree? Our main result gives a lower bound of n^{0.83}. We also give some upper bounds and bounds for general random tree growth models. This is based on joint work with Johannes Bäumler, Bas Lodewijks, James Martin, Emil Powierski, and Anirudh Sridhar.
September 26, 2024: Dmitry Krachun (Princeton)
A glimpse of universality in critical planar lattice models
Abstract: Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.
In this talk I will explain some ideas behind the recent result which proves rotational invariance of the FK-percolation model. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz.
Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.
October 3, 2024: Joshua Cape (UW-Madison)
A new random matrix: motivation, properties, and applications
In this talk, we introduce and study a new random matrix whose entries are dependent and discrete valued. This random matrix is motivated by problems in multivariate analysis and nonparametric statistics. We establish its asymptotic properties and provide comparisons to existing results for independent entry random matrix models. We then apply our results to two problems: (i) community detection, and (ii) principal submatrix localization. Based on joint work with Jonquil Z. Liao.
October 10, 2024: Midwest Probability Colloquium
N/A
October 17, 2024: Kihoon Seong (Cornell)
Gaussian fluctuations of focusing Φ^4 measure around the soliton manifold
I will explain the central limit theorem for the focusing Φ^4 measure in the infinite volume limit. The focusing Φ^4 measure, an invariant Gibbs measure for the nonlinear Schrödinger equation, was first studied by Lebowitz, Rose, and Speer (1988), and later extended by Bourgain (1994), Brydges and Slade (1996), and Carlen, Fröhlich, and Lebowitz (2016).
Rider previously showed that this measure is strongly concentrated around a family of minimizers of the associated Hamiltonian, known as the soliton manifold. In this talk, I will discuss the fluctuations around this soliton manifold. Specifically, we show that the scaled field under the focusing Φ^4 measure converges to white noise in the infinite volume limit, thus identifying the next-order fluctuations, as predicted by Rider.
This talk is based on joint work with Philippe Sosoe (Cornell).
October 24, 2024: Jacob Richey (Alfred Renyi Institute)
Stochastic abelian particle systems and self-organized criticality
Abstract: Activated random walk (ARW) is an 'abelian' particle system that conjecturally exhibits complex behaviors which were first described by physicists in the 1990s, namely self organized criticality and hyperuniformity. I will discuss recent results for ARW and the stochastic sandpile (a related model) on Z and other graphs, plus many open questions.
October 31, 2024: David Clancy (UW-Madison)
Likelihood landscape on a known phylogeny
Abstract: Over time, ancestral populations evolve to become separate species. We can represent this history as a tree with edge lengths where the leaves are the modern-day species. If we know the precise topology of the tree (i.e. the precise evolutionary relationship between all the species), then we can imagine traits (their presence or absence) being passed down according to a symmetric 2-state continuous-time Markov chain. The branch length becomes the probability a parent species has a trait while the child species does not. This length is unknown, but researchers have observed they can get pretty good estimates using maximum likelihood estimation and only the leaf data despite the fact that the number of critical points for the log-likelihood grows exponentially fast in the size of the tree. In this talk, I will discuss why this MLE approach works by showing that the population log-likelihood is strictly concave and smooth in a neighborhood around the true branch length parameters and the size.
This talk is based on joint work with Hanbaek Lyu, Sebastien Roch and Allan Sly.
November 7, 2024: Zoe Huang (UNC Chapel Hill)
Cutoff for Cayley graphs of nilpotent groups
Abstract: Abstract: We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly bounded. For some $k=k(n)$ such that $1\ll\log k \ll \log |G|$, we pick a random set of generators $S=S(n)$ by sampling $k$ elements $Z_1,\ldots,Z_k$ from $G$ uniformly at random with replacement, and set $S:=\{Z_j^{\pm 1}:1 \le j\le k \}$. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant $c>0$, depending only on the rank and the nilpotency class of $G$, such that for all symmetric sets of generators $S$ of size at most $ \frac{c\log |G|}{\log \log |G|}$, the spectral gap and the $\varepsilon$-mixing time of the simple random walk $X=(X_t)_{t\geq 0}$ on Cay$(G,S)$ are asymptotically the same as those of the projection of $X$ to the abelianization of $G$, given by $[G,G]X_t$. In particular, $X$ exhibits cutoff if and only if its projection does. Based on joint work with Jonathan Hermon.
November 14, 2024: Nabarun Deb (University of Chicago)
Mean-Field fluctuations in Ising models and posterior prediction intervals in low signal-to-noise ratio regimes
Ising models have become central in probability, statistics, and machine learning. They naturally appear in the posterior distribution of regression coefficients under the linear model $Y = X\beta + \epsilon$, where $\epsilon \sim N(0, \sigma^2 I_n)$. This talk explores fluctuations of specific linear statistics under the Ising model, with a focus on applications in Bayesian linear regression.
In the first part, we examine Ising models on "dense regular" graphs and characterize the limiting distribution of average magnetization across various temperature and magnetization regimes, extending previous results beyond the Curie-Weiss (complete graph) case. In the second part, we analyze posterior prediction intervals for linear statistics in low signal-to-noise ratio (SNR) scenarios, also known as the contiguity regime. Here, unlike standard Bernstein-von Mises results, the limiting distributions are highly sensitive to the choice of prior. We illustrate this dependency by presenting limiting laws under both correctly specified and misspecified priors.
This talk is based on joint work with Sumit Mukherjee and Seunghyun Li.
November 21, 2024: Reza Gheissari (Northwestern)
Wetting and pre-wetting in (2+1)D solid-on-solid interfaces
The (d+1)D-solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d\ge 2$, with zero-boundary conditions, at low temperatures the surface is localized about height $0$, but when constrained to take only non-negative values entropic repulsion pushes it to take typical heights of $O(\log n)$. I will describe the mechanism of entropic repulsion, and present results on how the picture changes when one introduces a competing force trying to keep the interface localized (either an external field or a reward for points where the height is exactly zero). Along the way, I will outline rich predictions for the shapes of level curves, and for metastability phenomena in the Glauber dynamics. Based on joint work with Eyal Lubetzky and Joseph Chen.
November 28, 2024: Thanksgiving
No seminar
December 5, 2024: Erik Bates (NC State)
Parisi formulas in multi-species and vector spin glass models
The expression "Parisi formula" refers to a variational formula postulated by Parisi in 1980 to give the limiting free energy of the Sherrington--Kirkpatrick (SK) spin glass. The SK model was originally conceived as a mean-field description for disordered magnetism, and has since become a mathematical prototype for frustrated disordered systems and high-complexity functions. In recent years, there has been an effort to extend the Parisi framework to various generalizations of the SK model, raising new physical questions met with fresh mathematical challenges. In this talk, I will share some developments in this evolving story. Based on joint works with Leila Sloman and Youngtak Sohn.