Schedule:
Friday, May 27th (Annenberg G21)
1:00pm  1:45pm

LouisPierre 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 logcorrelated 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

WeiKuo Chen (Minnesota)
A duality principle in meanfield 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 meanfield 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 pspin 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 nonplanar 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 KSAT 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 zeroone law for recurrence and transience of frog processes.
 Abstract
We provide sufficient conditions for the validity of a dichotomy, i.e. zeroone law, between recurrence and transience of frog models on a large class of nonrandom 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 quasitransitive Markov chains or by random walks in a common random environment including supercritical 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 zeroone 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 subcritical, supercritical 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 onedimensional random walk in a random environment  Abstract
For onedimensional 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 environmentdependent 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.)
Multiparticle diffusion limited aggregation  Abstract

3:15pm  3:45pm

Coffee Break

3:45pm  4:30pm

Gerard Ben Arous (NYU)
TBA  Abstract

4:450pm  5:30pm

Jack Hanson (CUNY)
Chemical distance in 2d critical percolation  Abstract
In critical twodimensional 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
pointtopoint and pointtobox distances.

6:00pm  (Harris Hall 108)

Reception

Sunday, May 29th (Lunt 105)
9:00am  9:30am

Breakfast

9:30am  10:15am

Alan Hammond (Berkeley)
Selfavoiding polygons and walks: counting, joining and closing.
 Abstract
Selfavoiding walk of length n on the integer lattice Z^d is the uniform measure on nearestneighbour 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 selfavoiding polygon. We investigate the numbers of selfavoiding walks, polygons, and in particular the "closing" probability that a length n selfavoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo DuminilCopin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on selfavoiding 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 twodimensional 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 IoffeVelenik 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.