What are the big problems in probability theory?

Question

Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long time, and currently has the classification of 4-manifolds. PDE theory has the Navier-Stokes equation to deal with.

So what are the big problems in probability theory and stochastic analysis?

I'm a grad student working in the field, but I can't name any major unsolved conjectures or open problems which are driving research. I've heard that stochastic Löwner evolutions are a big field of study these days, but I don't know what the conjectures or problems relating to them are.

Does anyone have any suggestions?

Answer 1

To my mind one of the biggest open problems in probability, in the sense of being a famous basic statement that we don't know how to solve, is to show that there is "no percolation at the critical point" (mentioned in particular in section 4.1 of Gordon Slade's contribution to the Princeton Companion to Mathematics). A capsule summary: write $\mathbb{Z}_{d,p}$ for the random subgraph of the nearest-neighbour $d$-dimensional integer lattice, obtained by independently keeping each edge with probability $p$. Then it is known that there exists a critical probability $p_c(d)$ (the percolation threshold}) such that for $p < p_c$, with probability one $\mathbb{Z}_{d,p}$ contains no infinite component, and for $p > p_c$, with probability one there exists an unique infinite component.

The conjecture is that with probability one, $\mathbb{Z}_{d,p_c(d)}$ contains no infinite component. The conjecture is known to be true when $d =2$ or $d \geq 19$.

Incidentally, one of the most effective ways we have of understanding percolation -- a technique known as the lace expansion, largely developed by Takeshi Hara and Gordon Slade -- is also one of the key tools for studying self-avoiding walks and a host of other random lattice models.

That article of Slade's is in fact full of intriguing conjectures in the area of critical phenomena, but the conjecture I just mentioned is probably the most famous of the lot.

Answer 2

Maybe the no1 problem of probability is to make rigorous what one finds in just about any textbook in statistical mechanics. In other words it is to put the predictions of Wilson's renormalization group theory on a rigorous footing. Many of the topics mentioned in this post are particular conjectures in this broader program.

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Update: A nice recent review on this topic by Gordon Slade can be found here.

Answer 3

Understand self-avoiding random walks, see http://gowers.wordpress.com/2010/08/22/icm2010-smirnov-laudatio/.

Answer 4

The normal distribution and the many places it occurs in mathematics and its application is a primary example of a universal phenomenon. Proving and understanding other universal phenomena in probability is of great importance. One example I like is to understand the distributions that came from random matrix theory and occur in various other places. One such distribution is the distribution of the largest eigenvalue of a random matrix discovered by Tracy and Widom.

Answer 5

Michel Talagrand has a number of open problems (with bounty) listed on his website. I haven't looked at them all, but knowing him, I guarantee you that they are very hard and quite important. These are motivated by his research directions, but unlike some fields, there's not one research direction and one set of open problems that dominate probability theory right now.

Answer 6

One major problem is extending the wonderful understanding of planar stochastic models to higher dimensions. So understanding 3,4-dimensional percolation, Ising Model, self avoiding walks, loop erased random walks and their scaling limits is a rather important problem.

Answer 7

To determine the limit shape of first passage percolation.

In the $n$-dimensional grid, start with a vertex colored black and all others colored white. Choose uniformly a bicolor edge (one black end, one white end) and color in black its white end. Continue this process forever.

The black part grows, and it is known that if we rescale it so that it has constant diameter, it converges to a convex shape. What we do not know is what the shape is.

Answer 8

The lack of a so-called big problem in probability theory seems to suggest the richness of the subject itself. One of the most fascinating subfields is the determination of convergence rate of finite state space Markov chains. Many convergence problem even on finite groups have exhausted current analytic techniques. For instance, intuitions from coupon collector's problem suggests that the random adjacent transposition walk exhibits cutoff in total variation convergence to the uniform measure on the symmetric group, and the upper and lower bounds gap is only a factor of 2. There are many tools one could employ to study such problems, such as representation theory and discretized version of inequalities from PDE theory, which makes the solutions very creative.

Answer 9

David Aldous has a list of open problems on his website, though they look like personal favorites rather than "big" questions. You might look at the problems Aldous labels as "Type 2:We have a precise mathematical problem, but we do not see any plausible outline for a potential proof."

Chapter 23 of the recent monograph Markov Chains and Mixing Times is a list of open problems. Again, though, I cannot say which of these are "big."

Answer 10

Maybe the 1917 Cantelli conjecture? If $f$ is a positive function on real numbers, if $X$ and $Z$ are $N(0,1)$ independent rv such that $X+f(X)Z$ is normal, prove that $f$ is a constant ae.

Answer 11

You can also have a look at the list of open problems on Michael Aizenman's homepage:

http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/

These are very important for (mathematical) physics, and several fall in the realm of probability theory (in particular: Soft phases in 2D O(N) models, and Spin glass).

Answer 12

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following:

Let $(X_n,n\in\Bbb N)$ be a strictly stationary $\phi$-mixing sequence, for which $\mathbb E(X_0^2)<\infty$ and $\operatorname{Var}(S_n)\to +\infty$. Then $S_n:=\sum_{j=1}^nX_j$ is asymptotically normally distributed.

$\phi$-mixing coefficents are defined as $$\phi_X(n):=\sup(|\mu(B\mid A)-\mu(B)|, A\in\mathcal F^m, B\in \mathcal F_{m+n},m\in\Bbb N ),$$ where $\mathcal F^m$ and $\mathcal F_{m+n}$ are the $\sigma$-algebras generated by the $X_j$, $j\leqslant m$ (respectively $j\geqslant m+n)$, and $\phi$-mixing means that $\phi_X(n)\to 0$.

It was posed in Ibragimov and Linnik paper in 1965.

Peligrad showed the result holds with the assumption $\liminf_{n\to +\infty}n^{-1}\operatorname{Var}(S_n)>0$. It also holds when $\mathbb E\lvert X_0\rvert^{2+\delta}$ is finite for some positive $\delta$ (Ibragimov, I think).

Answer 13

Forming tools for handling random surfaces and proving an universal central limit theorem for them given minimal conditions (think usual CLT):

a)The Gaussian Free Field (GFF) has shown up as the limiting universal object for many random surfaces (in KPZ 2+1, random tilings, random matrix theory Ginibre ensembles cf Borodin's and Kenyon's work)

b)Schramm-Loewner evolutions (SLE) have shown up as the limiting interfaces for families of statistical models.

c)Finally, merging the above two pictures since SLEs can be coupled to GFF (cf Sheffield).

Another powerful result would be showing the equivalence of Random planar maps and Liouville quantum gravity (LQG) (a promising approach by Miller and Sheffield). This is because it happens that statistical models become easier to handle over these random surfaces (Kazanov-Ising model, LERW-Duplantier).