# What are the big problems in probability theory?

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?

• Though this question is imperfect, I vote to keep it open. As a frequent consumer of probability theory I find it interesting and useful. Aug 31 '10 at 6:14
• I feel that the answers, while nice, leave large areas of probability untouched. Sep 1 '10 at 7:57
• As much as I love maths and their open problems, I don't think the word sexy applies to them Dec 6 '20 at 19:43

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.

• I agree that this conjecture (referred to as the dying percolation conjecture) is a great open problem. It is especially challenging in dimensions 3,4, and 5 where the Hara Slade results do not hold. Aug 30 '10 at 21:56

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.

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

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

• Understanding SAW is certainly one of the biggest outstanding problems in probability theory. Nonetheless, it's premature to select it as The Answer within hours of posting your question. Aug 30 '10 at 18:50
• Since it is not CW, this means there is one, unique, answer. So this must be it! Aug 30 '10 at 21:39

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.

• Just to provide a link for more information, here is the Wikipedia article on T-W distribution: en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution Sep 24 '10 at 14:01
• and T. Tao has recently blogged about similar ideas: terrytao.wordpress.com/2010/09/14/… Sep 24 '10 at 15:31
• Other such phenomena that seem less than completely explained are the Newcomb–Benford law, Pareto's law, and Zipf's law. $\qquad$ Nov 9 at 19:36

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.

• I like the use of the word "very" to mean "probably extremely" (from what little I've tried to read of Talagrand's stuff) Aug 30 '10 at 21:55

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.

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.

• What Benoît has described is the Richardson growth model, which has limiting shape equal to that of first-passage percolation with i.i.d. exponential passage times. What is most fascinating to me is that the limiting shape is not known for any distribution of i.i.d. passage times. There are related models for which the limiting shape is known (e.g. last-passage percolation, Euclidean FPP, FPP with stationary and ergodic passage times), but the i.i.d. case has resisted all attack. Aug 30 '10 at 19:01
• Another variation (the general Richardson model) is to choose some $p$, and for each boundary edge to color its white end black with probability $p$. The limit shape is not known except obviously for $p=1$, and when $p\to 0$ the limit shape converges to the limit shape of first passage percolation. An interesting fact is known, though: if $p$ is close enough to $1$, then the limit shape is not strictly convex. Aug 31 '10 at 8:15

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.

• Number theory has several big problems but is also a very rich subject (not all work in number theory is directed towards the Riemann hypothesis or the Birch and Swinnerton-Dyer conjecture), so the lack of a big problem in probability does not really point to the subject's richness. Feb 2 '11 at 17:30
• I suppose if you couple it with the fact so many people work in it, then richness does become a corollary. Feb 2 '11 at 19:55
• I wasn't intending to suggest probability is not a rich subject, but I don't buy the argument that the lack of a very prominent unsolved question or program in an area is in some way a sign that the area is rich. Seems kind of after-the-fact justification to me. Apr 15 '14 at 21:17
• Modern mathematicians never cease to amaze me. Just found out that the random adjacent transposition walk has been shown to exhibit cutoff at the Wilson lower bound back in 2016, after reading one of my advisor Persi's latest paper: arxiv.org/pdf/1309.3873.pdf Apr 8 '18 at 5:23
• @John Jiang there has also been progress on the "limit profile" of the random transposition random walk and also there has been a fairly comprehensive study of the random transposition walk on the quantum permutation group! Nov 9 at 20:33

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."

• In an earlier version (stat.berkeley.edu/~aldous/Research/problems.ps) of that list of open problems, Aldous states that 'they are not intended to be "representative" or "the most important" ... of all open problems in probability. The majority are (I think) my own invention and have not been discussed extensively elsewhere'. That having been said, I really enjoy Aldous' list and find many of his open problems dangerously fun to think about. Aug 30 '10 at 19:44
• "Simulating the scaling exponent in Euclidean random TSP" is at the end of Aldous's list -- is that still open fifteen years later? stat.berkeley.edu/~aldous/Research/OP/simTSP.html Nov 9 at 20:06

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.

• What kind of information is there out there about the history of this problem? Feb 3 '11 at 1:56
• Victor Kleptsyn and Aline Kurtzmann claim to give a counterexample (front.math.ucdavis.edu/1202.2250 ). Feb 13 '12 at 19:39
• The counterexample is now published at jstor.org/stable/24520369 Nov 9 at 19:56

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).

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).

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).