Combinatorial probability

I think about problems in mathematics with probabilistic flavor. The motivation for most of my recent work is from network science, statistical physics, and pure combinatorics, but I have also worked on projects related to machine learning, statistical inference, translation surfaces, and dynamical systems. I often use computer simulations to help form conjectures, and recently, AI has become a part of my day to day thinking process. Below are short descriptions and relevant materials for my research on activated random walk, shifts of finite type and pattern problems, random matrices, combinatorial stochastic processes, and random walks.

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Animated odometer heat maps for point-source activated random walk in subcritical, near-critical, and supercritical sleeping backgrounds — **Subcritical***ρ = 0.45* **Critical window***ρ ≈ ρ_c(ℤ²) ≈ 0.66* **Supercritical***ρ = 0.80* Point-source ARW with λ = 1 on three 320 × 320 boxes with absorbing boundaries. Color records the logarithmic odometer.

Five broad areas, one scroll at a time

Interacting particles · critical phenomena

Activated random walk

In the 1980s, Per Bak, Chao Tang, and Kurt Wiesenfeld coined and popularized the term self-organized criticality (SOC), a phenomenon they observed in systems like avalanches, forest fires, and stock markets. In SOC systems, energy slowly builds and is released in shocks whose sizes follow a statistical power law. ARW is a stochastic particle model of SOC introduced in the early 2000s.

Simulations show that ARW has a self-organized critical limiting state, but a full mathematical description and proof is still missing. While most models of SOC are beyond our current mathematical techniques, significant progress has been made recently on ARW due to newly discovered special properties that are intrinsic to the system: in addition to the abelian property, ARW is believed to exhibit a new kind of monotonicity and negative association.

Preprint · 2026Law of large numbers for ARW on villages↗ Preprint · 2026Local density of activated random walk↗ Preprint · 2026Explosivity in one-dimensional ARW↗ Preprint · 2025The odometer in subcritical ARW↗ PreprintStochastic sandpile on ℤ↗ CMP · 2023Activated random walk on ℤ↗ AIHP · 2019Activated random walk on a cycle↗ SlidesRényi Institute Seminar talk↗

Animated local-density heat map beside a graph of retained particle density over time for driven-dissipative activated random walk — **Figure 01**Driven-dissipative ARW in a square with absorbing boundary; darker blue means greater local particle density.

Symbolic dynamics · pattern avoidance

Shifts of finite type & patterns

Consider colorings of a Cayley graph, for example the 2D lattice, and fix a finite family of forbidden patterns, i.e. finite subsets with a fixed coloring. Any such family gives rise to the shift of finite type (SFT) consisting of all colorings of the full space which never have any translated copy of a forbidden pattern. What does a typical configuration look like, and what statistical properties does it have?

Many interesting questions arise already in one dimension. When a single pattern is forbidden, the entropy of the resulting SFT can be described by a simple probabilistic formula, and conjecturally, a similar formula can be extended to arbitrary 1D SFTs. In higher dimensions, computing entropy or establishing mixing is much harder, but some partial results can be obtained by comparing different SFTs with explicit combinatorial maps.

Preprint · 2026Letter frequency with one forbidden word↗ DCDS · 2026Forbidding a single pattern↗ EJC · 2025Word length, bias and bijections in Penney’s ante↗ Living notesPatterns in SFTs↗

Square lattice sample at beta minus one, where monochromatic two-by-two squares are discouraged — **Figure 02**Top row: monochromatic 2 × 2 squares receive weight e^β. Bottom row: 111 is forbidden vertically and horizontally, and 1s receive weight e^β.

Square lattice sample at beta one point two five, where monochromatic two-by-two squares are rewarded — **Figure 02**Top row: monochromatic 2 × 2 squares receive weight e^β. Bottom row: 111 is forbidden vertically and horizontally, and 1s receive weight e^β.

Random matrices · statistical inference

Random matrices

A recurring theme in the study of large finite graphs is to describe hidden geometric features. Roughly speaking, a random graph ensemble has geometry if it can be roughly embedded in a low-dimensional metric space. For Wishart matrices—random graphs embedded in a sphere—one can characterize a transition between detectable geometry, if the dimension of the sphere is relatively low compared to the size of the graph, and noise, if the dimension is relatively high.

Random intersection graphs share essentially the same transition. This suggests a universal phenomenon: can one characterize a broad class of random graphs that lose geometry at the same threshold? These random graph ensembles play a central role in the theory of machine learning and data science.

JTP · 2019A smooth transition from Wishart to GOE↗

In preparationWhen intersection graphs lose geometry

Dimension ratio d/n³

Figure 03GOE and Wishart are symmetric, so their eigenvalues lie on the real line; vertical jitter separates the samples. Move d/n³ through the transition from distinguishable matrix laws to nearly matching ones.

Random algorithms · percolation · random sets

Combinatorial stochastic processes

I am eternally drawn to problems about random combinatorial objects: Markov chains, fragmentation/coagulation processes, renewal processes, random number theory, and random algorithms. Often, simple rules can give rise to complex systems which go beyond the scope of modern probabilistic techniques or reveal new phenomena. When theory fails, computer simulations can quickly come to the rescue, and even sometimes suggest a path forward.

Figure 04 begins with IID points in either a square or a disk and repeatedly deletes the point whose Voronoi cell has the smallest area. In each row, one election runs beside a heat-map histogram of the final survivor. Perhaps surprisingly, the winner distribution is not maximized at the center, but along an annulus.

Living notesCombinatorial processes↗ NotesRandom sum constructions↗ Short noteGeometric coupons↗ JAP · 2022Intersections of random sets↗ Undergraduate thesisCounting clusters on a finite grid↗ REU paperPieces in percolated hexagonal grids↗

Animated Voronoi deletion election in a square shrinking from eight hundred candidates to one winner — **Figure 04**Minimum-cell Voronoi deletion on two domains. Each animated run ends at one survivor; each neighboring histogram combines 1,200 independent elections.

Heat map of winner locations from repeated Voronoi deletion elections in a square — **Figure 04**Minimum-cell Voronoi deletion on two domains. Each animated run ends at one survivor; each neighboring histogram combines 1,200 independent elections.

Inverse problems · perturbations · interacting paths

Random walks

Random walks are the bread and butter of my research. Here is a tantalizing inverse problem related to simple random walk: given only the trace of a 2D random walk, i.e. the set of visited sites, can one make a good guess about where it began? Recent work of Goenka, Keevash, and Przybyłowski proves that no method succeeds with positive constant probability in dimensions one or two, while finitely many guesses suffice in dimensions at least five; dimensions three and four remain open.

Below are some of my other writings on random walk systems, including coalescing/annihilating walks, permuted walks, and an expository note containing some original calculations related to collisions of multiple random walks.

EJP · 2025Diffusion-limited annihilating-coalescing systems↗ EJP · 2024Random walks on regular trees cannot be slowed down↗ SlidesFinding a source from its trace↗ IEEE · 2020Rumor source detection↗ Survey & calculationsRandom walk collisions↗ Recent result · 2026Source localisation in simple random walks↗

Figure 05Occupation times from a freshly generated million-step planar simple random walk. Brightness records repeated visits; the star reveals the exact starting site.