Workshop on percolation, spin glasses and random media

Schedule:

Friday, May 27th (Annenberg G21)

1:00pm - 1:45pm	Louis-Pierre Arguin (CUNY) Maxima of the characteristic polynomial of random unitary matrices - Abstract A recent conjecture of Fyodorov, Hiary & Keating (FHK) states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of log-correlated Gaussian fields. In this talk, we will highlight the connections between the two problems. We will outline the proof of the conjecture for the leading order of the maximum, as well as the free energy and the entropy of high points. We will also discuss the connections with the FHK conjecture for the maximum of the Riemann zeta function on the critical line. This is based on joint works with D. Belius (NYU), P. Bourgade (NYU), and A. Harper (Cambridge).
2:00pm - 2:45pm	Wei-Kuo Chen (Minnesota) A duality principle in mean-field spin glasses - Abstract Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. In particular, it is famously known that the limiting free energies in many mean-field spin glasses can be computed through Parisi's variational representations. In this talk, we will present a general duality principle for the limiting free energy. We will discuss how such structure appears naturally in the random energy model and the mixed p-spin model and explain its connection with the Parisi formula. Joint work with Antonio Auffinger.
2:45pm - 3:30pm	Coffee Break
3:30pm - 4:15pm	Michael Aizenman (Princeton) On Pfaffian relations in planar and non-planar two dimensional models - Abstract
4:30pm - 5:15pm	David Gamarnik (MIT) Finding a Large Submatrix of a Random Matrix, and the Overlap Gap Property - Abstract Many problems in random combinatorial structures exhibit an apparent gap between the existential results and algorithmically achievable results, though no formal complexity theoretic hardness of these problems is known. Examples include the problem of proper coloring of a random graph, finding a largest independent set of a graph, random K-SAT problem, and many others. In our talk we consider a new example of such a gap for the problem of finding a submatrix which achieves the largest average value in a given random matrix. We will consider some known and a new algorithm for this problem, all of which produce a matrix with average value constant factor away from the globally optimal one. Then, motivated by the theory of spin glasses, we consider the overlap structure of pairs of matrices achieving a certain average value, and show that it undergoes a certain connectivity phase transition just above the value achievable by the best known algorithm. We conjecture that the onset of this overlap gap property marks the onset of the algorithmic hardness for this problem and in fact we conjecture that this is the case for most randomly generated optimization problems. Joint work with Quan Li (MIT)

Saturday, May 28th (Lunt 105)

9:30am - 10:00am

Breakfast

10:00am - 10:45am

Elena Kosygina (CUNY)
A zero-one law for recurrence and transience of frog processes. - Abstract

We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of frog models on a large class of non-random and on some random graphs. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on Zd. We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on Zd. Its proof uses the general zero-one law. This is a joint work with Martin Zerner (Universitaet Tuebingen, Germany).

11:00am - 11:45am

Augusto Texeira (IMPA)
Sharpness of the phase transition for continuum percolation on R^2 - Abstract

In this talk we will discuss the phase transition of random radii Poisson Boolean percolation: around each point of a planar Poisson Process, we draw a disc of random radius, independently for each point. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter. We will then explain several results on the sub-critical, super-critical and critical phases of this process, that resemble what happens for Bernoulli independent percolation The techniques we present in this talk are general and can be applied to other models such as the Poisson Voronoi and Confetti percolation. This talk is based on a joint work with Daniel Ahlberg and Vincent Tassion.

1:30pm - 2:15pm

Jon Peterson (Purdue)
Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment - Abstract

For one-dimensional random walks in a random environment with positive limiting speed $v_0>0$ and with environments having both local drifts to the right and to the left, it is known that the large deviation probabilities of moving at a speed $v$ in $(0,v_0)$ which is slower than the typical speed decays slower than exponentially fast. In this talk I will consider precise asymptotics of these slowdown probabilities under the quenched measure. We will show that these quenched probabilities decay like $e^{-C_n(\omega) n^{-\gamma}}$ for some fixed $\gamma \in (0,1)$ and for some environment-dependent sequence $C_n(omega)$ which oscillates between $0$ and $\infty$. This confirms a conjecture of Gantert and Zeitouni. This talk is based on joint work with Sung Won Ahn.

2:30pm - 3:15pm

Vladas Sidoravicius (NYU and NYU - Sh.)
Multi-particle diffusion limited aggregation - Abstract

TBA

3:15pm - 3:45pm

Coffee Break

3:45pm - 4:30pm

Gerard Ben Arous (NYU)
TBA - Abstract

TBA

4:450pm - 5:30pm

Jack Hanson (CUNY)
Chemical distance in 2d critical percolation - Abstract

In critical two-dimensional Bernoulli percolation, 1/2 of the edges of the graph Z^2 are erased independently. The resulting graph has connected components and "holes" appearing on all scales. As a result, the chemical (graph) distance inside large connected components is conjectured to grow superlinearly in the Euclidean distance, and some results in this direction are known. For instance, the shortest crossing of the box [-n, n]^2 has length S_n > n^{1 + \epsilon} with high probability, and is no longer than the unique lowest crossing, whose length L_n is known to scale as n^{4/3 + o(1)}. Kesten and Zhang asked whether S_n = o(L_n); we will discuss recent work which gives an affirmative answer to this question, as well as some results on point-to-point and point-to-box distances.

6:00pm - (Harris Hall 108)

Reception

Sunday, May 29th (Lunt 105)

9:00am - 9:30am

Breakfast

9:30am - 10:15am

Alan Hammond (Berkeley)
Self-avoiding polygons and walks: counting, joining and closing. - Abstract

Self-avoiding walk of length n on the integer lattice Z^d is the uniform measure on nearest-neighbour walks in Z^d that begin at the origin and are of length n. If such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. The result of doing so is a self-avoiding polygon. We investigate the numbers of self-avoiding walks, polygons, and in particular the "closing" probability that a length n self-avoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo Duminil-Copin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on self-avoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most n^{-1/2 + o(1)} in any dimension at least two. Developing a quite different method of polygon joining employed by Madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. We further make use of the snake method and polygon joining technique at once to prove upper bounds on the closing probability below n^{-1/2} in the two-dimensional setting.

10:30am - 11:15am

Naoki Kubota (Tokyo)
Concentration inequalities for the simple random walk in unbounded nonnegative potentials - Abstract

n this talk, we consider the simple random walk in i.i.d. nonnegative potentials on the multidimensional cubic lattice, and study the cost paid by the simple random walk for traveling from the origin to a remote location in a landscape of potentials. In particular, the focus of this talk is the concentration inequality for the travel cost in unbounded nonnegative potentials. It has already been proved by Ioffe--Velenik and Sodin for potentials with bounded and strictly positive support, and the main result in this talk extends a part of their works to the case where potentials are unbounded and nonnegative.

11:30am - 12:15pm

Chris Hoffman (Washington)
Geodesics in First Passage Percolation - Abstract

I will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This work will show that in some sense the work of Damron and Hanson about the existence of geodesics is optimal. This is joint work with Gerandy Brito and Daniel Ahlberg.

Organizers

The organizers are Antonio Auffinger and Elton Hsu.

Acknowledgements

This meeting is partially supported by a grant from the National Science Foundation to the probability group at Northwestern University and by the Northwestern Mathematics Department as part of the 2015/2016 emphasis year in probability theory.

9:30am - 10:00am	Breakfast
10:00am - 10:45am	Elena Kosygina (CUNY) A zero-one law for recurrence and transience of frog processes. - Abstract We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of frog models on a large class of non-random and on some random graphs. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on Zd. We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on Zd. Its proof uses the general zero-one law. This is a joint work with Martin Zerner (Universitaet Tuebingen, Germany).
11:00am - 11:45am	Augusto Texeira (IMPA) Sharpness of the phase transition for continuum percolation on R^2 - Abstract In this talk we will discuss the phase transition of random radii Poisson Boolean percolation: around each point of a planar Poisson Process, we draw a disc of random radius, independently for each point. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter. We will then explain several results on the sub-critical, super-critical and critical phases of this process, that resemble what happens for Bernoulli independent percolation The techniques we present in this talk are general and can be applied to other models such as the Poisson Voronoi and Confetti percolation. This talk is based on a joint work with Daniel Ahlberg and Vincent Tassion.
1:30pm - 2:15pm	Jon Peterson (Purdue) Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment - Abstract For one-dimensional random walks in a random environment with positive limiting speed $v_0>0$ and with environments having both local drifts to the right and to the left, it is known that the large deviation probabilities of moving at a speed $v$ in $(0,v_0)$ which is slower than the typical speed decays slower than exponentially fast. In this talk I will consider precise asymptotics of these slowdown probabilities under the quenched measure. We will show that these quenched probabilities decay like $e^{-C_n(\omega) n^{-\gamma}}$ for some fixed $\gamma \in (0,1)$ and for some environment-dependent sequence $C_n(omega)$ which oscillates between $0$ and $\infty$. This confirms a conjecture of Gantert and Zeitouni. This talk is based on joint work with Sung Won Ahn.
2:30pm - 3:15pm	Vladas Sidoravicius (NYU and NYU - Sh.) Multi-particle diffusion limited aggregation - Abstract TBA
3:15pm - 3:45pm	Coffee Break
3:45pm - 4:30pm	Gerard Ben Arous (NYU) TBA - Abstract TBA
4:450pm - 5:30pm	Jack Hanson (CUNY) Chemical distance in 2d critical percolation - Abstract In critical two-dimensional Bernoulli percolation, 1/2 of the edges of the graph Z^2 are erased independently. The resulting graph has connected components and "holes" appearing on all scales. As a result, the chemical (graph) distance inside large connected components is conjectured to grow superlinearly in the Euclidean distance, and some results in this direction are known. For instance, the shortest crossing of the box [-n, n]^2 has length S_n > n^{1 + \epsilon} with high probability, and is no longer than the unique lowest crossing, whose length L_n is known to scale as n^{4/3 + o(1)}. Kesten and Zhang asked whether S_n = o(L_n); we will discuss recent work which gives an affirmative answer to this question, as well as some results on point-to-point and point-to-box distances.
6:00pm - (Harris Hall 108)	Reception

9:00am - 9:30am	Breakfast
9:30am - 10:15am	Alan Hammond (Berkeley) Self-avoiding polygons and walks: counting, joining and closing. - Abstract Self-avoiding walk of length n on the integer lattice Z^d is the uniform measure on nearest-neighbour walks in Z^d that begin at the origin and are of length n. If such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. The result of doing so is a self-avoiding polygon. We investigate the numbers of self-avoiding walks, polygons, and in particular the "closing" probability that a length n self-avoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo Duminil-Copin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on self-avoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most n^{-1/2 + o(1)} in any dimension at least two. Developing a quite different method of polygon joining employed by Madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. We further make use of the snake method and polygon joining technique at once to prove upper bounds on the closing probability below n^{-1/2} in the two-dimensional setting.
10:30am - 11:15am	Naoki Kubota (Tokyo) Concentration inequalities for the simple random walk in unbounded nonnegative potentials - Abstract n this talk, we consider the simple random walk in i.i.d. nonnegative potentials on the multidimensional cubic lattice, and study the cost paid by the simple random walk for traveling from the origin to a remote location in a landscape of potentials. In particular, the focus of this talk is the concentration inequality for the travel cost in unbounded nonnegative potentials. It has already been proved by Ioffe--Velenik and Sodin for potentials with bounded and strictly positive support, and the main result in this talk extends a part of their works to the case where potentials are unbounded and nonnegative.
11:30am - 12:15pm	Chris Hoffman (Washington) Geodesics in First Passage Percolation - Abstract I will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This work will show that in some sense the work of Damron and Hanson about the existence of geodesics is optimal. This is joint work with Gerandy Brito and Daniel Ahlberg.