The seminar covers a wide range of topics in pure and applied probability and in mathematical physics.
Unless otherwise noted, the talks take place on **Fridays, 11am-12noon**, in room **WWH 512**. See directions to the Courant Institute.

Lisa Hartung and Thomas Leblé help organizing, in behalf of the Probability/Mathematical physics group.
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Old webpage.

The random-cluster model is a dependent percolation model where the weight of a configuration is proportional to q to the power of the number of connected components. It is highly related to the ferromagnetic q-Potts model, where every vertex is assigned one of q colors, and monochromatic neighbors are encouraged. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever $q > 4$ - i.e. there are multiple Gibbs measures at criticality. We provide a rigorous proof of this claim. Like Baxter, our proof uses the correspondence between the above models and the six-vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter's formula for the correlation length of the models at criticality. This is joint work with Hugo Duminil-Copin, Maxime Gangebin, Ioan Manolescu, and Vincent Tassion.

Consider the problem of recovering $v \in \mathbb S^{N-1}$ from an i.i.d. Gaussian $k$-tensor spiked by the rank-one tensor $v^{\otimes k}$. The likelihood function for this is an example of a complex high-dimensional landscape. It is information-theoretically possible to detect the spike at order 1 signal-to-noise ratios, but if $k>2$ it is expected that the signal-to-noise ratio needs to diverge in $N$ to efficiently recover $v$. We seek to understand the mechanisms for, and obstructions to, such planted signal recovery in high dimensions, via locally optimizing algorithms.
We study recovery thresholds for a family of ``vanilla" algorithms known as Langevin dynamics, on general
landscapes consisting of a planted non-linear signal function perturbed by isotropic Gaussian noise.
We propose a mechanism for success/failure of recovery via such algorithms in terms of the curvature of the planted signal.

Joint work with G. Ben Arous and A. Jagannath.

Consider a perturbed lattice ${v+Y_v}$ obtained by adding IID d-dimensional Gaussian variables ${Y_v}$ to the lattice points in $Z^d$. Suppose that one point, say $Y_0$, is removed from this perturbed lattice; is it possible for an observer, who sees just the remaining points, to detect that a point is missing? In one and two dimensions, the answer is positive: the two point processes (before and after $Y_0$ is removed) can be distinguished using smooth statistics, analogously to work of Sodin and Tsirelson (2004) on zeros of Gaussian analytic functions. (cf. Holroyd and Soo (2011)). The situation in higher dimensions is more delicate; our solution depends on a game-theoretic idea, in one direction, and on the unpredictable paths constructed by Benjamini, Pemantle and the speaker (1998), in the other.(Joint work with Allan Sly).

What can one say about a graph from multiple (short) random walk trajectories on it? In this talk we consider algorithms that only "see" walk trajectories and the degrees along the way.

We will show that the number of vertices, edges and mixing time can be all estimated with a number of RW steps that is sublinear in the size of the graph and in its mixing or relaxation time. Our bounds on the number of RW steps are optimal up to constant or polylog factors. We also argue that such algorithms cannot "know when to stop", and discuss additional conditions that circumvent this limitation. To prove our results, we rely on bounds for random walk intersections by Peres, Sauerwald, Sousi and Stauffer.

Joint with Anna Ben-Hamou (Sorbonne) and Yuval Peres (MSR).