Probability and Mathematical Physics Seminar

Synopsis:

Critical models are, almost by definition, supposed to feature a slow
decay of correlations for local observables while retaining some mixing
even for macroscopic quantities. A strong version of the latter property
is that changing boundary conditions cannot have a singular (in the
measure theoretic sense) effect on the model away from the boundary,
even asymptotically. I will present a proof of this result for the dimer
model as an illustration of new techniques in the study of the dimer
model involving a reformulation of the problem in terms of uniform
spanning tree. Joint work with Antoine Bannier.

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Synopsis:

Parabolic Anderson model (PAM) is one of the prototypical frameworks for modelling conduction of electrons in crystals filled with defects. Intermittency of the peaks of the PAM is one of the widely studied topics in the last few decades and it holds close ties with the phenomenon of Anderson localization.  We show that the peaks of the PAM in dimension 2 and 3 are macroscopically multifractal. More precisely, we prove that the spatial peaks of the PAM have infinitely many distinct values and we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) of those peaks. As a byproduct, we obtain the exact spatial asymptotics of the solution of the PAM. We also study the spatio-temporal peaks of the PAM and show their macroscopic multifractality. Some of the major tools used in our proof techniques include paracontrolled calculus and tail probabilities of the largest point in the spectrum of the Anderson Hamiltonian. This talk is based on a joint work with Jaeyun Yi.

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Synopsis:

The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.

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Synopsis:

There is an intense debate in finance whether volatility in asset prices is rough or not. In non-finance language, this question can be rephrased as follows: Is it possible to determine the regularity of a process $\sigma_t$ if only its stochastic integral $\int_0^t \sigma_s dB_s$ can be observed? In this talk, I will show that the answer is “yes” and I will discuss a CLT result that can be used to construct an estimator with the best possible rate of convergence in a minimax sense. On the way, I will also explain how similar probabilistic tools can be used to solve statistical questions arising in turbulence physics and stochastic PDE models.

This talk is based on joint work with Marc Hoffmann (Paris Dauphine), Yanghui Liu (Baruch College), Mathieu Rosenbaum and Grégoire Szymanski (both Ecole Polytechnique).
 

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Synopsis:

The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The second meeting of Spring 2023 will feature:

• Lingfu Zhang (UC Berkeley)
  Title: Random lozenge tiling at cusps and the Pearcey process

  Abstract: It has been known since Cohn-Kenyon-Propp (2000) that uniformly random tiling by lozenges exhibits frozen and disordered regions, which are separated by the 'arctic curve'. For a generic simply connected polygonal domain, the microscopic statistics are widely predicted to be universal, being one of (1) discrete sine process inside the disordered region (2) Airy line ensemble around a smooth point of the curve (3) Pearcey process around a cusp of the curve (4) GUE corner process around a tangent point of the curve. These statistics were proved years ago for special domains, using exact formulas; as for universality, much progress was made more recently. In this talk, I will present a proof of the universality of (3), the remaining open case. Our approach is via a refined comparison between tiling and non-intersecting random walks, for which a new universality result of the Pearcey process is also proved. This is joint work with Jiaoyang Huang and Fan Yang.

• Bjoern Bringmann (IAS)
  Title: Invariant Gibbs measures for the three-dimensional cubic nonlinear wave equation.
  
  Abstract: In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ^4_3-model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ^4_3-model by Hairer ’14 and Hairer-Matetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toy-models. We also connect our theorem with classical and recent developments in constructive QFT, dispersive PDEs, and stochastic PDEs. In the second half of this talk, we give a non-technical overview of the proof. As part of this overview, we first introduce a caloric representation of the Gibbs measure, which leads to an interplay of both parabolic and hyperbolic theories. Then, we discuss our para-controlled Ansatz and a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.

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Synopsis:

We consider a continuous time simple random walk on a subset of the square lattice with wired boundary conditions: The walk transitions at unit edge rate on the graph obtained from the lattice closure of the subset by contracting the boundary into one vertex. We study the cover time of such walk, namely the time it takes for the walk to visit all vertices in the graph. Taking a sequence of subsets obtained as scaled lattice versions of a nice planar domain, we show that the square root of the cover time normalized by the size of the subset, is tight around (1/\pi) \log N - (1/4\pi) \log \log N, where N is the scale parameter. This proves an analog, in the wired case, of a conjecture by Bramson and Zeitouni. The proof is based on comparison with the extremal landscape of the discrete Gaussian free field.

Joint work with Marek Biskup and Santiago Saglietti.

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Notes:

A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. In our work, we rely on a smaller set of tools including Hall’s matching theorem and a double dimer swapping operation. In this talk, I will explain how to formulate the large deviations principle in 3D, show simulations, give some of the key ideas of our proofs. Time permitting, I will briefly describe a (seemingly simple) open problem which highlights one of the ways that three dimensions is different from two.

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Synopsis:

A basic goal for random graph matching is to recover the vertex correspondence between two correlated graphs from an observation of these two unlabeled graphs. Random graph matching is an important and active topic in combinatorial statistics: on the one hand, it arises from various applied fields such as social network analysis, computer vision, computational biology and natural language processing; on the other hand, there is also a deep and rich theory that is of interest to researchers in statistics, probability, combinatorics, optimization, algorithms and complexity theory.

Recently, extensive efforts have been devoted to the study for matching two correlated Erdős–Rényi graphs, which is arguably the most classic model for graph matching. In this talk, we will review some recent progress on this front, with emphasis on the intriguing phenomenon on (the presumed) information-computation gap. In particular, we will discuss progress on efficient algorithms thanks to the collective efforts from the community. We will also point out some important future directions, including developing robust algorithms that rely on minimal assumptions on graph models and developing efficient algorithms for more realistic random graph models. 

This is based on joint works with Hang Du, Shuyang Gong, Zhangsong Li, Zongming Ma, Yihong Wu and Jiaming Xu. 

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Synopsis:

In this talk I will present results on two new point processes on the unit circle. At first sight, these point processes look similar to the well-studied circular-\beta-ensemble, but in fact there are very different: instead of featuring the 2d Coulomb interactions ~|e^{i\theta_j} - e^{i\theta_k}|^\beta, they are characterized by the non-local interactions ~|e^{i\theta_j} - e^{-i\theta_k}|^\beta and ~|e^{i\theta_j} + e^{i\theta_k}|^\beta, respectively. These point processes can be seen as \beta-ensembles of a new kind with a lot more randomness than in the classical \beta-ensembles. I will mostly be discussing the smooth linear statistics.

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Synopsis:

The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The first meeting of Spring 2023 will feature:

• Reza Gheissari (Northwestern University)
  Title: Cutoff in the Glauber dynamics for the Gaussian free field
  
  Abstract:
  The Gaussian free field (GFF) is a canonical model of random surfaces, generalizing the Brownian bridge to two dimensions. It arises naturally as the stationary solution to the stochastic heat equation with additive noise (SHE), and together the SHE and GFF are expected to be the universal scaling limit of many random surface evolutions arising in lattice statistical physics. We consider the mixing time (time to converge to stationarity, when started out of equilibrium) for the pre-limiting object, the discrete Gaussian free field (DGFF) evolving under the Glauber dynamics. We establish that on a box of side-length $n$ in $\mathbb Z^2$, the Glauber dynamics for the DGFF exhibits the cutoff phenomenon, mixing exactly at time $\frac{2}{\pi^2} n^2 \log n$. Based on joint work with S. Ganguly.
   
• Xin Sun (University of Pennsylvania)
  Title: Random surface, planar lattice model, and conformal field theory
  
  Abstract:
  Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via  another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on random surfaces and planar lattice models, including the law of the random modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical planar percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT.  Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.

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Synopsis:

Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an alpha-stable law; such distributions have infinite variance when alpha is less than 2. Due to the ubiquity of heavy-tailed randomness, these models have been broadly applied in physics, finance, and statistics. When the entries have infinite mean, Lévy matrices are predicted to exhibit a phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the physical context for this conjecture, and describe a result establishing it for values of alpha close to zero and one. This is joint work with Amol Aggarwal and Charles Bordenave.

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Synopsis:

A permuton is a probability measure on $[0,1]^2$ whose two coordinate marginals are Lebesgue measure. Permutons describe the large-scale geometry of random permutations. I will discuss a geometric construction of a certain class of random permutons using a pair of random space-filling curves called Schramm-Loewner evolution (SLE) and a random measure arising from Liouville quantum gravity (LQG). This class includes the limits of various types of random pattern-avoiding permutations as well as the conjectural limit of meandric permutations (permutations arising from a simple loop which crosses a line a specified number of times). I will then discuss some results about random permutations which can be proven using SLE and LQG, concerning, e.g., the length of the longest increasing subsequence and the fractal dimension of the support.

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Synopsis:

Two distinct regimes appear in large DNNs, depending on the variance of the parameters at initialization. For large initializations, the initial parameters lie inside a narrow valley of global minima and gradient flow converges very fast to a nearby local minimum, never approaching any saddle. The infinite width dynamics in this regime are approximately linear and are described by the Neural Tangent Kernel (NTK). In contrast for small initialization, we observe in linear networks a Saddle-to-Saddle regime, where gradient flow visits the neighborhoods of a sequence of saddles, each corresponding to linear maps of increasing rank. This leads to an implicit bias towards learning low rank linear maps. Similar properties are observed in non-linear networks.

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Synopsis:

In the interchange process on a graph , distinguished particles are placed on the vertices of  with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation  is formed for any time . One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles.

We prove the existence of infinite cycles in the interchange process on  for all dimensions  and all large , establishing a conjecture of Bálint Tóth from 1993 in these dimensions.

In our proof, we study a self-interacting random walk called the cyclic time random walk.  Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.

This is a joint work with Allan Sly.

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