These problems are mostly original, and (of course) reflect my
personal taste.
They are definitely **not** intended as
*the most important open problems in Probability*,
and I do not follow the most active
current research areas.
Historically I have emphasised problems that seem conceptually somewhat novel, rather than
"natural next step" problems suggested by a specific known result.
But upon retirement (2018) I am adding some of the latter.

I categorize problems on a conceptual-technical spectrum. That is

**Type 1**We have a question in words, but we don't know how to state a precise mathematical problem which both seems capable of solution and whose solution would be interesting.**Type 2**We have a precise mathematical problem, but we do not see any plausible outline for a potential proof.**Type 3**We have a precise mathematical problem, and a plausible outline for a potential proof, but we cannot carry through the technical details.

- (0.5) Martingale, for practical purposes.
- (0.7) Congestion in networks: SOC, HOT and the percolation-fragmentation phase transition.
- (1.5) Data compression for sparse labeled graphs.
- (1.6) Mixing times for coagulation-fragmentation processes.
- (1.7) The low-density limit of coalescing branching random walk.
- (1.8) Bacon and eggs: task scheduling into batches.
- (1.9) Metropolis on Cayley graphs.
- (2.0) Spectral gap of Bayesian graph Laplacian.
- (2.0) Phase transitions for general epidemic models on networks.
- (2.0) Shortest routes in random proximity networks.
- (2.1) Mixing times for the branch-rotation chain on cladograms.
- (2.2) Random Eulerian circuits.
- (2.3) The stretch-length tradeoff in spatial networks.
- (2.6) Largest common substructures in probabilistic combinatorics.
- (2.6) Percolation and empires.
- (2.8) A spatial model of city growth and formation.
- (2.9) Scale-invariant random spatial networks.
- (3.0) Topological properties of a random partition of the plane.
- (3.0) Local Weak Limits and Unimodularity.
- (3.0) The online MST of the randomly-weighted complete graph.
- (3.1) A one-dimensional drift-jump particle process.
- (3.2) A conjectured compactification of some finite reversible Markov chains.
- (3.4) A discrete Hammersley process as an extreme case of oriented percolation flows.

In a different realm, I really really really want someone to do this simulation. Of course, it's not easy!

Also, pages 9-12 of Lecture 3 in these Local weak convergence of random graphs and networks lectures give a "recursive distributional equation" I would like someone to solve numerically.

- How many Brownian particles escape when you control with total drift = 1?
- WLLN for first-passage percolation on finite graphs..
- Spectral gap for the interchange (exclusion) process on a finite graph.
- Independent sets in sparse random graphs
- Power laws and killed branching random walk.
- Mixing time for a Gibbs sampler on the simplex.
- Scaling window for percolation of averages in the mean-field setting.
- Performance of the Metropolis algorithm on a disordered tree.
- Coding invariant processes on infinite trees.
- Wright-Fisher diffusions with negative mutation rate!
- An unusually-coupled simple random walk.
- Greedy tour length and the greedy spatial service algorithm.
- What is the max-entropy win-probability martingale?.

Itai Benjamini

Krzysztof
Burdzy

Tom Liggett

Michel Talagrand