1 2const publications = [ 3 { 4 id: "manhattan-lattice-transient", 5 title: "The randomly oriented Manhattan lattice in 2D is transient", 6 bibtexTitle: "The randomly oriented {M}anhattan lattice in {2D} is transient", 7 thumbnail: "images/manhattan_lattice.svg", 8 thumbnailAnimation: "manhattan-lattice", 9 altText: "A walker follows randomly oriented horizontal and vertical lines and drifts away from its starting point.", 10 authors: [ 11 { name: "Ahmed Bou-Rabee" }, 12 { name: "Yuval Peres", url: "https://yuvalperes.com/" } 13 ], 14 links: [ 15 { type: "arXiv", url: "https://arxiv.org/abs/2609.07621" }, 16 { type: "PDF", url: "https://arxiv.org/pdf/2609.07621" }, 17 { type: "Lean", url: "https://github.com/nitromannitol/Manhattan-Transience" } 18 ], 19 abstract: 'Independently orient each horizontal and vertical line of Z<sup>2</sup> by a fair coin. A walker chooses one of the two lines through its current position with equal probability and takes one step in the direction of that line. <a href="https://doi.org/10.1016/0167-2789(89)90207-8">Redner (1989)</a> introduced this walk as a model of transport in an isotropic random velocity field and predicted that its root-mean-square displacement grows like n<sup>2/3</sup>. We prove that the walk is transient almost surely. The proof is a short variational argument.', 20 year: 2026 21 }, 22 { 23 id: "orrw-range-exponent", 24 title: "Once-reinforced random walk on Z<sup>d</sup> has range exponent at least d/(d+1)", 25 bibtexTitle: "Once-reinforced random walk on {$\\mathbb{Z}^d$} has range exponent at least $d/(d+1)$", 26 thumbnail: "images/orrw_range.svg", 27 thumbnailAnimation: "orrw-range", 28 altText: "A once-reinforced walk retraces the trail it has already made, and its range grows slowly.", 29 authors: [ 30 { name: "Ahmed Bou-Rabee" }, 31 { name: "Yuval Peres", url: "https://yuvalperes.com/" } 32 ], 33 status: "submitted", 34 links: [ 35 { type: "arXiv", url: "https://arxiv.org/abs/2610.00090" }, 36 { type: "PDF", url: "https://arxiv.org/pdf/2610.00090" }, 37 { type: "Lean", url: "https://github.com/nitromannitol/ORRW-Lower-Bound" } 38 ], 39 abstract: 'In once-reinforced random walk, every edge starts with weight one and is assigned weight β ≥ 1 after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. Davis (1990) introduced the model on Z. For sufficiently large reinforcement β, the expected range in the first n steps is predicted to have order n<sup>d/(d+1)</sup> on Z<sup>d</sup>. We prove this prediction as a lower bound in every dimension d ≥ 2 and for every reinforcement parameter β.', 40 year: 2026 41 }, 42 { 43 id: "sharpness-critical-scaling-parking", 44 title: "Sharpness and critical scaling of parking", 45 thumbnail: "images/parking.svg", 46 thumbnailAnimation: "parking", 47 altText: "Cars performing square-lattice random walks and occupying distinct free parking spots.", 48 authors: [ 49 { name: "Ahmed Bou-Rabee" }, 50 { name: "Christoforos Panagiotis", url: "https://sites.google.com/view/christoforos-panagiotis/" } 51 ], 52 status: "submitted", 53 links: [ 54 { type: "arXiv", url: "https://arxiv.org/abs/2609.02820" }, 55 { type: "Gallery", url: "gallery/#parking" }, 56 { type: "Lean", url: "https://github.com/nitromannitol/Parking-Sharpness" } 57 ], 58 abstract: 'In the parking model, each site of the d-dimensional lattice independently starts with one car with probability p or one parking spot with probability 1−p. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime p = 1/2, the expected number of visits to a site in n rounds is of order n<sup>(4−d)/4</sup> for d ≤ 3 and log n for d ≥ 4. We also prove that in the subcritical regime p ∈ (0, 1/2), the parking-time tail is bounded above and below by stretched exponentials with exponent d/(d+2). As p ↑ 1/2, we also determine the divergence of the expected total number of visits to a site: its order is (1−2p)<sup>−
583</sup>, (1−2p)<sup>−1</sup> and (1−2p)<sup>−1/3</sup> in dimensions one, two and three, respectively, and log(1/(1−2p)) in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of <a href="https://doi.org/10.1007/s11118-008-9104-6">Levine and Peres</a> plus a martingale-type term. These results answer questions posed by <a href="https://doi.org/10.1214/18-AAP1443">Damron, Gravner, Junge, Lyu and Sivakoff</a>.', 59 year: 2026 60 }, 61 { 62 id: "quantitative-explosion-percolation-divisible-sandpile", 63 title: "Quantitative explosion and percolation of the divisible sandpile", 64 thumbnail: "images/divisible_percolation.svg", 65 thumbnailAnimation: "divisible-percolation", 66 altText: "Parallel divisible-sandpile topplings joining into a percolating nearest-neighbor component near critical density.", 67 authors: [ 68 { name: "Ahmed Bou-Rabee" }, 69 { name: "Christoforos Panagiotis", url: "https://sites.google.com/view/christoforos-panagiotis/" } 70 ], 71 status: "submitted", 72 links: [ 73 { type: "arXiv", url: "https://arxiv.org/abs/2609.02829" }, 74 { type: "Gallery", url: "gallery/#divisible-percolation" }, 75 { type: "Lean", url: "https://github.com/nitromannitol/Divisible-Sandpile-Percolation" } 76 ], 77 abstract: 'The divisible sandpile on Z<sup>d</sup> starts from i.i.d. masses at each site, and, in each discrete time step, a site with mass above one keeps one unit and sends the excess equally to its neighbors. <a href="https://doi.org/10.1007/s00023-015-0433-x">Levine, Murugan, Peres and Ugurcan</a> showed that at mean one this process explodes, with every site emitting infinite mass. We show that the mass emitted from a site by time t is of order t<sup>(4−d)/4</sup> for d ≤ 3, of order log t for d = 4, and a tail-dependent, divergent rate for d ≥ 5. We further show that the mass emitted, after diffusive rescaling, converges to a Brownian optimal-stopping value for d ≤ 3 and to tail-dependent, weighted membrane fields for d ≥ 5, while at the critical dimension d = 4, after superdiffusive rescaling, it converges to the membrane model. Using these estimates, we prove that, for every d ≥ 2, the set of sites that topple contains an infinite component at some mean below one, hence it has a non-trivial percolation phase transition. This answers a variant of a question of <a href="https://doi.org/10.1214/08-AOP415">Fey, Meester and Redig</a>. The proof adapts ideas from the theory of level-set percolation of strongly correlated Gaussian fields.', 78 year: 2026 79 }, 80 { 81 id: "eulerian-walkers", 82 title: "Eulerian walkers on Z<sup>2</sup> have range exponent 2/3", 83 bibtexTitle: "Eulerian walkers on {$\\mathbb{Z}^2$} have range exponent 2/3", 84 thumbnail: "images/eulerian_walkers.png", 85 thumbnailAnimation: "rotor-walk", 86 altText: "An Eulerian walker rotates lattice arrows and traverses a range colored by circuit.", 87 authors: [ 88 { name: "Ahmed Bou-Rabee" }, 89 { name: "Yuval Peres", url: "https://yuvalperes.com/" } 90 ], 91 links: [ 92 { type: "arXiv", url: "https://arxiv.org/abs/2608.23545" }, 93 { type: "Lean", url: "https://github.com/nitromannitol/rotor-23" } 94 ], 95 abstract: 'In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by 90° and steps in the new direction. <a href="https://arxiv.org/abs/cond-mat/9611019">Priezzhev, Dhar, Dhar, and Krishnamurthy (1996)</a> introduced this as a model of self-organized criticality and conjectured that, for independent uniform initial directions, the region explored in the first t steps has radius of order t<sup>1/3</sup>. We establish this conjecture and further show that the walker visits every lattice site infinitely often, and that the region it has visited by time t, rescaled by t<sup>1/3</sup>, converges to a convex body.', 96 year: 2026 97 }, 98 { 99 id: "einstein-relation", 100 title: "Quantitative Einstein relation for reversible diffusions in a random environment", 101 thumbnail: "gallery/plates/p-einstein-thumbnail.svg", 102 thumbnailAnimation: "einstein-relation", 103 altText: "Paired diffusion histories: time runs left to right, with blue unforced paths and gold paths under a small constant force.", 104 authors: [ 105 { name: "Ahmed Bou-Rabee" }, 106 { name: "Ruizhe Xu", url: "https://arxiv.org/search/?query=%22Ruizhe+Xu%22&searchtype=author&abstracts=show&order=-announced_date_first&size=50" } 107 ], 108 links: [ 109 { type: "arXiv", url: "https://arxiv.org/abs/2605.26082" } 110 ], 111 abstract: 'A Brownian particle in a random environment, when pushed by a small constant force, acquires a deterministic asymptotic velocity. The <a href="https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)">Einstein relation</a> predicts that the velocity-to-force ratio converges to the diffusivity of the unforced particle as the force vanishes — equating the particle\'s linear response to its equilibrium fluctuations. This is one of the most basic instances of the <a href="https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem">fluctuation–dissipation principle</a> in statistical physics. For reversible diffusions in stationary random environments, <a href="https://arxiv.org/abs/1005.5665">Gantert, Mathieu, and Piatnitski</a> proved this identity as a qualitative limit. We prove a quantitative version: the mobility error decays as an explicit power of the force magnitude λ, in a quenched sense. The key observation is that under a force of magnitude λ, the drift λt and the diffusive spread √t balance at spatial scale λ<sup>-1</sup>; at this scale the problem becomes one of <a href="https://arxiv.org/abs/2210.06488">quantitative homogenization</a>. Our rate is not expected to be optimal: we conjecture that the sharp rate is λ in d ≥ 3 and λ√(log(1/λ)) in d = 2, reflecting the <a href="https://arxiv.org/abs/1705.05300">growth of the corrector</a> in the two regimes. It would be particularly interesting to prove this conjecture, which we expect will require understanding how the steady state of the environment seen from the particle expands to second order in λ.', 112 year: 2026 113 }, 114 { 115 id: "critical-long-range-rcm", 116 title: "Quantitative homogenization for the critical long-range random conductance model", 117 thumbnail: "images/critical_long_range_rw.png", 118 thumbnailAnimation: "long-range-walk", 119 altText: "A critical long-range random walk combining local motion with occasional long jumps.", 120 authors: [ 121 { name: "Ahmed Bou-Rabee" }, 122 { name: "Paul Dario", url: "https://pauldario.pages.math.cnrs.fr/webpage-of-paul-dario/" } 123 ], 124 links: [ 125 { type: "arXiv", url: "https://arxiv.org/abs/2604.21143" } 126 ], 127 abstract: 'Consider a random walk on Z<sup>d</sup> that jumps from x to y at rate a(x,y) |x-y|<sup>-(d+α)</sup>, where the conductances a(x,y) are i.i.d. and uniformly elliptic. The exponent α controls the large-scale behavior: for α < 2, <a href="https://arxiv.org/abs/1805.04344">the walk converges to an α-stable process</a>; for α > 2, <a href="https://arxiv.org/abs/2004.01971">it converges to Brownian motion</a> under the usual diffusive scaling. At the critical exponent α = 2 the second moment of the jump kernel diverges logarithmically, and the walk sits between these two regimes. We prove quantitative homogenization of the associated nonlocal elliptic equation to the Laplacian at the rate 1/√|log ε|, and, as a consequence, deduce quenched convergence of the random walk to Brownian motion under the anomalous √(t log t) scaling. A feature of the critical regime is that the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order â in contrast to the classical homogenization setting. Our proof proceeds by a two-scale expansion, adapting the <a href="https://arxiv.org/abs/1705.05300">quantitative homogenization program of Armstrong, Kuusi, and Mourrat</a> to the nonlocal critical setting via a new Poincaré inequality for the critical kernel.', 128 year: 2026 129 }, 130 { 131 id: "gff-idla-cylinder", 132 title: "Gaussian fluctuations for Internal DLA on cylinders", 133 thumbnail: "images/idla_cylinder.png", 134 thumbnailAnimation: "idla-cylinder", 135 altText: "Internal DLA growing upward from the base of a periodic cylinder.", 136 authors: [ 137 { name: "Ahmed Bou-Rabee" }, 138 { name: "Vittoria Silvestri", url: "https://vittoriasilvestri.wordpress.com/" }, 139 { name: "Ariel Yadin", url: "https://www.math.bgu.ac.il/en/people/users/yadina" } 140 ], 141 links: [ 142 { type: "arXiv", url: "https://arxiv.org/abs/2604.21142" } 143 ], 144 abstract: 'Internal DLA is a random growth model whose limit shape and fluctuations are well understood on the lattice Z<sup>d</sup> and on the cylinder Z<sub>N</sub> × Z. On Z<sup>d</sup>, <a href="https://arxiv.org/abs/1101.0596">David Jerison, Lionel Levine, and Scott Sheffield</a> showed that the average fluctuations converge to a variant of the Gaussian free field (GFF); in a <a href="https://arxiv.org/abs/1310.5063">companion paper</a> the same authors considered IDLA on the cylinder Z<sub>N</sub> × Z and proved that space-time averages of the fluctuations converge to the ordinary GFF. In this note we establish universality of the GFF limit across cylinder geometries: for any vertex-transitive base graph V<sub>N</sub> whose rescaled low spectrum converges and whose spectral gap vanishes, the average fluctuations of IDLA on V<sub>N</sub> × Z converge to a GFF on the corresponding half-cylinder. The prototypical case is V<sub>N</sub> = Z<sub>n</sub><sup>d</sup>, where the limit lives on T<sup>d</sup> × R<sub>+</sub>
144. Along the way we present an improved maximal-fluctuation bound, which is of independent interest and which implies a shape theorem for IDLA on V<sub>N</sub> × Z for any vertex-transitive base. Our proof extends the martingale strategy of Jerison, Levine, and Sheffield to arbitrary vertex-transitive bases, the main new input being an EfronâStein variance inequality that exploits the abelian property of IDLA.', 145 year: 2026 146 }, 147 { 148 id: "divisible-sandpile-rwrs", 149 title: "Divisible sandpiles via random walks in random scenery", 150 thumbnail: "images/divisible_sandpile.png", 151 thumbnailAnimation: "sandpile-rwrs", 152 altText: "A divisible-sandpile odometer evolving over a fixed random reward field.", 153 authors: [ 154 { name: "Ahmed Bou-Rabee" }, 155 { name: "Yuval Peres", url: "https://yuvalperes.com/" }, 156 { name: "Ecaterina Sava-Huss", url: "https://www.math.tugraz.at/~sava/" } 157 ], 158 links: [ 159 { type: "arXiv", url: "https://arxiv.org/abs/2604.13968" }, 160 { type: "Lean", url: "https://github.com/nitromannitol/Divisible-Sandpile-RWRS" } 161 ], 162 abstract: 'The divisible sandpile, <a href="https://arxiv.org/abs/0704.0688">introduced by Lionel Levine and Yuval Peres</a>, is a continuous-mass version of the abelian sandpile: a vertex with mass greater than 1 distributes the excess equally among its neighbors. Starting from i.i.d. initial masses with mean μ, the pile either stabilizes everywhere or explodes. <a href="https://arxiv.org/abs/1501.07258">Levine, Murugan, Peres, and Ugurcan</a> proved a sharp mean-based phase transition on vertex-transitive graphs: stabilization when μ < 1, explosion when μ > 1, and explosion at μ = 1 when the variance is positive and finite. A key ingredient in their proof is that toppling conserves the mean mass on transitive graphs. This conservation fails on general graphs, and, surprisingly, a single heavy-tailed fluctuation can drive explosion even when μ < 1. In this paper, we extend their phase transition to all infinite bounded-degree graphs, and show that our conditions are nearly sharp. Instead of conservation of mass, we recast stabilization as an optimal stopping problem for random walk in random scenery. Along the way, we strengthen the critical case on Z<sup>d</sup>: at μ = 1, explosion holds whenever the centered law is symmetric and nonzero, with no finite-variance hypothesis.', 163 year: 2026 164 }, 165 { 166 id: "algebraic-sd", 167 title: "Superdiffusion and anomalous regularization in self-similar random incompressible flows", 168 thumbnail: "images/algebraic-sd.png", 169 thumbnailAnimation: "algebraic-superdiffusion", 170 altText: "A jagged Brownian path wandering through a finite random incompressible flow with several spatial scales.", 171 authors: [ 172 { name: "Ahmed Bou-Rabee" }, 173 { name: "Scott Armstrong", url: "https://www.scottnarmstrong.com" }, 174 { name: "Tuomo Kuusi", url: "https://sites.google.com/site/tuomokuusimath/home" } 175 ], 176 links: [ 177 { type: "arXiv", url: "https://arxiv.org/abs/2601.22142" }, 178 { type: "Blog", url: "https://www.scottnarmstrong.com/2026/01/superdiffusivity-anomalous-regularization/" }, 179 { type: "Lean", url: "https://github.com/scottnarmstrong/Superdiffusion" } 180 ], 181abstract: 'A Brownian particle advected by an incompressible random drift undergoes enhanced diffusion. When the drift has long-range correlationsâspecifically, when the stream matrix has positive Hurst exponent γ > 0âphysicists <a href="references/sdphys1.pdf">predicted</a> <a href="references/sdphys2.pdf">since</a> <a href="references/sdphys3.pdf">the</a> <a href="references/sdphys4.pdf">1980s</a> that the particle should be superdiffusive, with mean squared displacement growing like t<sup>2/(2-γ)</sup>. We prove this prediction in the perturbative regime γ ⪠1. In <a href="https://arxiv.org/abs/2404.01115">previous work</a>, we treated the critical case γ = 0 and proved convergence to Brownian motion under superdiffusive scaling; here the power-law growth is too fast for Gaussian behavior, and we show the particle has tails inconsistent with any Brownian limit. We also prove anomalous regularization: solutions of the associated elliptic equation are Hölder continuous with exponent 1 - Cγ<sup>1/2</sup>, uniformly in the molecular diffusivity. See this <a href="https://www.scottnarmstrong.com/2026/01/superdiffusivity-anomalous-regularization/">blog post</a> for more.',
182 year: 2026 183 }, 184 { 185 id: "random-walk-sphere-packings", 186 title: "Random walk on sphere packings and Delaunay triangulations in arbitrary dimension", 187 authors: [ 188 { name: "Ahmed Bou-Rabee" }, 189 { name: "Ewain Gwynne", url: "https://math.uchicago.edu/~ewain/" } 190 ], 191 journal: "Proceedings of the London Mathematical Society", 192 volume: "131", 193 pages: "e70084", 194 links: [ 195 { type: "Journal", url: "https://doi.org/10.1112/plms.70084" }, 196 { type: "arXiv", url: "https://arxiv.org/abs/2405.11673" } 197 ], 198abstract: 'Random walks on graphs approximating Euclidean space are known to converge to Brownian motion. We introduce a family of graphs, orthogonal tiling graphs, flexible enough to approximate irregular manifolds in arbitrary dimension, for which we prove that the trace of random walk converges to the trace of Brownian motion. This family includes Voronoi tessellations of Gaussian multiplicative chaos in arbitrary dimension, and so our result implies a higher-dimensional generalization of <a href="https://arxiv.org/abs/1807.07511">Gwynne, Miller, and Sheffield (2022)</a>. Our proof proceeds by showing convergence of a finite volume scheme that induces a non-uniformly elliptic operator on the graph. One of our main contributions is an L<sup>2</sup>-L<sup>â</sup> estimate for this operator. In the special case of 2D, our method also leads to a short proof of <a href="https://arxiv.org/abs/1906.01613">Gurel-Gurevich, Jerison, and Nachmias (2020)</a>, which in turn generalized earlier work by <a href="references/dubjeko1999.pdf">Dubejko (1999)</a>, <a href="https://arxiv.org/abs/0810.2188">Chelkak-Smirnov (2011)</a>, <a href="https://arxiv.org/abs/1110.6737">Skopenkov (2013)</a>, and <a href="https://arxiv.org/abs/1511.01209">Werness (2015)</a>. Orthogonal tiling graphs in 2D include isoradial graphs, which were used by <a href="https://arxiv.org/abs/0910.2045">
198Chelkak and Smirnov (2012)</a> to establish the universality of critical Ising models. We believe orthogonal tiling graphs could help in studying this model and others in d > 2.', 199 thumbnail: "images/rw_sphere_packing.png", 200 thumbnailAnimation: "sphere-packing", 201 altText: "A conductance-weighted random walk on an irregular planar orthogonal tiling.", 202 year: 2025 203 }, 204 { 205 id: "superdiffusive-clt-brownian-particle", 206 title: "Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift", 207 authors: [ 208 { name: "Ahmed Bou-Rabee" }, 209 { name: "Scott Armstrong", url: "https://www.scottnarmstrong.com" }, 210 { name: "Tuomo Kuusi", url: "https://sites.google.com/site/tuomokuusimath/home" } 211 ], 212 journal: "Inventiones Mathematicae", 213 status: "to appear", 214 links: [ 215 { type: "arXiv", url: "https://arxiv.org/abs/2404.01115" }, 216 { type: "Quanta", url: "https://www.quantamagazine.org/new-superdiffusion-proof-probes-the-mysterious-math-of-turbulence-20250516/" }, 217 { type: "Blog", url: "https://www.scottnarmstrong.com/2024/04/superdiffusive-CLT/" } 218 219 ], 220abstract: 'A Brownian particle advected by an incompressible vector field undergoes enhanced diffusion. The extent of this enhancement depends on the correlation structure of the field. Above a critical correlation threshold, the particle is superdiffusive, with mean squared displacement scaling algebraically in time (t<sup>α</sup>, α > 1). Below the threshold, diffusion is enhanced, but the particle retains the usual diffusive scaling (α = 1). In the critical case studied here, the mean squared displacement has order tâlog(t), so the typical displacement is ât·(log t)<sup>1/4</sup>. More precisely, for almost every frozen environment, |log ε<sup>2</sup>|<sup>â1/4</sup> εX<sub>t/ε<sup>2</sup></sub> converges in law to â(2c<sub>*</sub>)W<sub>t</sub>. This rate was predicted using heuristic renormalization-group arguments beginning in the 1980s. Our proof makes the renormalization argument rigorous using stochastic homogenization and gives a quenched Brownian path limit under the critical superdiffusive scaling. See this <a href="https://www.scottnarmstrong.com/2024/04/superdiffusive-clt/">blog post</a> and <a href="https://www.quantamagazine.org/new-superdiffusion-proof-probes-the-mysterious-math-of-turbulence-20250516/">Quanta article</a> for more.', 221 thumbnail: "images/sd.png", 222 thumbnailAnimation: "critical-superdiffusion", 223 altText: "A Brownian particle leaving a growing, irregular trail across the streamlines of a finite random incompressible flow.", 224 year: 2024 225 }, 226 { 227 id: "unique-continuation-planar-graphs", 228 title: "Unique continuation on planar graphs", 229 authors: [ 230 { name: "Ahmed Bou-Rabee" }, 231 { name: "Bill Cooperman", url: "https://cprmn.org/" }, 232 { name: "Shirshendu Ganguly", url: "https://www.stat.berkeley.edu/~sganguly/" } 233 ], 234 journal: "Discrete Analysis", 235 volume: "2025:16", 236 pages: "12 pp.", 237 links: [ 238 { type: "Journal", url: "https://doi.org/10.19086/da.144015" }, 239 { type: "arXiv", url: "https://arxiv.org/abs/2309.13728" }, 240 { type: "Lean", url: "https://github.com/nitromannitol/Unique-Continuation-Planar" } 241 ], 242 abstract: 'Bounded harmonic functions are constant. However, there are non-constant harmonic functions on R<sup>d</sup> that are bounded on all but a tiny fraction of space (see, <a href="https://link.springer.com/book/10.1007/978-3-642-61983-0">[Ch 3, Problems 158-160]</a>). These functions have no immediate discrete analogue, and, it turns out, cannot exist on periodic planar graphs. We prove that for any periodic planar graph G, there exists a constant α(G) such that any discrete harmonic function which is bounded on a (1 - α(G))-density of the graph is constant. A key aspect of the argument is a new unique continuation result for periodic planar graphs. Our proof is based on the maximum principle and elementary geometric arguments. It generalizes and simplifies earlier work by <a href="https://arxiv.org/abs/1712.07902">Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova, and Mikhail Sodin</a>, who established this result on the square lattice using techniques specific to that setting. The ideas in Buhovsky-Logunov-Malinnikova-Sodin were later applied by <a href="https://arxiv.org/abs/1809.09041">Charles Smart and Jian Ding</a> to establish localization near the edge for the Anderson-Bernoulli model on the square lattice.', 243 thumbnail: "images/unique_continuation.png", 244 thumbnailAnimation: "unique-continuation", 245 altText: "A candidate nonzero component inside a planar graph ball is enclosed by a zero cycle and then disappears from the inner ball, as required by unique continuation.", 246 year: 2025 247 }, 248 { 249 id: "rigidity-harmonic-functions", 250 title: "Rigidity of harmonic functions on the supercritical percolation cluster", 251 authors: [ 252 { name: "Ahmed Bou-Rabee" }, 253 { name: "Bill Cooperman", url: "https://cprmn.org/" }, 254 { name: "Paul Dario", url: "https://pauldario.pages.math.cnrs.fr/webpage-of-paul-dario/" } 255 ], 256 journal: "Transactions of the American Mathematical Society", 257 volume: "378(06)", 258 pages: "3823-3896", 259 links: [ 260 { type: "arXiv", url: "https://arxiv.org/abs/2303.04736" } 261 ], 262 abstract: 'H. A. Heilbronn <a href="references/heilbronn.pdf">showed</a> in 1949 that the dimension of the space of discrete harmonic polynomials on Z<sup>d</sup> coincides with that of R<sup>d</sup>. This theorem holds for any periodic lattice and was recently extended by <a href="https://arxiv.org/abs/1609.09431">Scott Armstrong and Paul Dario</a> to the supercritical percolation cluster, where their proof used the fact that, on large scales, the cluster resembles R<sup>d</sup>. However, in models such as the <a href="https://arxiv.org/abs/1109.0449?context=math">Ising model</a>, the <a href="https://arxiv.org/abs/1012.4809">Abelian sandpile</a>, and <a href="https://arxiv.org/abs/1508.04284">Anderson localization</a>
262, small-scale changes in the graph induce qualitatively distinct global behavior. Motivated by this, we prove three theorems on the supercritical cluster that do not hold on Z<sup>d</sup>. Using quantitative homogenization, we show the absence of non-constant Lipschitz harmonic functions and the absence of non-constant integer-valued harmonic functions of at most linear growth. In dimension 2, we also show that a function harmonic outside a finite set, with integer-valued graph Laplacian and uniform decay faster than 1/|x|, must have finite support. We expect these results and our techniques will help in understanding other models on the cluster. (The final section includes many conjectures and partial results regarding the Abelian sandpile on the cluster.)', 263 thumbnail: "images/lipschitz-percolation-harmonic.png", 264 thumbnailAnimation: "percolation-harmonic", 265 altText: "The width-four resistance ladder whose growing denominators force integer-valued harmonic functions to be constant.", 266 year: 2025 267 }, 268 { 269 id: "internal-dla-mated-crt-maps", 270 title: "Internal DLA on mated-CRT maps", 271 authors: [ 272 { name: "Ahmed Bou-Rabee" }, 273 { name: "Ewain Gwynne", url: "https://math.uchicago.edu/~ewain/" } 274 ], 275 journal: "Annals of Probability", 276 volume: "52(6)", 277 pages: "2173-2237", 278 links: [ 279 { type: "arXiv", url: "https://arxiv.org/abs/2211.04891" } 280 ], 281 abstract: 'We prove the convergence of internal diffusion limited aggregation on a family of random planar maps, mated-CRT maps, to the LQG harmonic balls constructed in <a href="https://arxiv.org/abs/2208.11795">our previous work</a>. The proof combines an idea of <a href="https://arxiv.org/abs/1111.0486">Hugo Duminil-Copin, Cyrille Lucas, Ariel Yadin, and Amir Yehudayoff</a> together with inputs from LQG theory. Mated-CRT maps are currently the only random planar maps for which the <a href="https://arxiv.org/abs/2003.10320">convergence of random walk to Liouville Brownian motion</a> has been established, and this fact plays a crucial role in our proof. It would be particularly interesting to determine the law governing the fluctuations around the limit shape.', 282 thumbnail: "images/idla_matedcrt.png", 283 thumbnailAnimation: "mated-crt-idla", 284 altText: "Internal DLA cells settling from the root of a mated-CRT map and forming an LQG harmonic ball.", 285 year: 2024 286 }, 287 { 288 id: "harmonic-balls-liouville-quantum-gravity", 289 title: "Harmonic balls in Liouville quantum gravity", 290 authors: [ 291 { name: "Ahmed Bou-Rabee" }, 292 { name: "Ewain Gwynne", url: "https://math.uchicago.edu/~ewain/" } 293 ], 294 journal: "Proceedings of the London Mathematical Society", 295 volume: "130", 296 pages: "e70018", 297 year: 2025, 298 links: [ 299 { type: "arXiv", url: "https://arxiv.org/abs/2208.11795" } 300 ], 301 abstract: 'We prove the well-posedness of Hele-Shaw flow on Liouville Quantum Gravity (LQG) surfaces and then use this to construct LQG harmonic balls, domains that satisfy the mean value property on LQG surfaces. LQG harmonic balls are the conjectured scaling limit of internal diffusion-limited aggregation on random planar maps. Almost surely, for Lebesgue-almost every mass, they are neither <a href = "https://arxiv.org/abs/2109.01252">LQG metric balls</a> nor Lipschitz domains; for every mass, the boundaries of their complementary components are Jordan curves. There are many interesting properties of LQG harmonic balls to explore, such as their Hausdorff dimension and the structure of their induced geodesics.', 302 thumbnail: "images/harmonic_ball.png", 303 thumbnailAnimation: "harmonic-ball", 304 altText: "An LQG harmonic ball evolving through a nonuniform random area field.", 305 }, 306 { 307 id: "integer-superharmonic-matrices", 308 title: "Integer superharmonic matrices on the F-lattice", 309 authors: [ 310 { name: "Ahmed Bou-Rabee" } 311 ], 312 journal: "Advances in Mathematics", 313 volume: "436", 314 pages: "109400", 315 year: 2024, 316 links: [ 317 { type: "Journal", url: "https://doi.org/10.1016/j.aim.2023.109400" }, 318 { type: "arXiv", url: "https://arxiv.org/abs/2110.07556" }, 319 { type: "Appendix", url: "manuscripts/appendix.pdf" }, 320 { type: "Code", url: "https://github.com/nitromannitol/f-lattice-recursion" } 321 ], 322abstract: 'The scaling limit of the Abelian sandpile is a fully nonlinear elliptic PDE with a delicate lattice dependence. This dependence is expressed through the quadratic growths of integer-valued, lattice-superharmonic functions (see <a href="https://arxiv.org/pdf/1611.00411">Section 6 of this survey</a> for an exposition). <a href="https://annals.math.princeton.edu/2017/186-1/p01">Lionel Levine, Wesley Pegden, and Charles Smart showed</a> that these functions on Z<sup>2</sup> are characterized by an Apollonian packing of the plane. While their proof was specific to Z<sup>2</sup>, with some effort it can likely be adapted to the triangular, hexagonal, and trihexagonal lattices, where the Apollonian packing is replaced by other circle packings (see these <a href="https://www.math.cmu.edu/~wes/sandgallery.html">simulations by Pegden</a>). The F-lattice presents a distinct situation: its integer superharmonic matrices are classified by a re
322cursion through rational points on a hyperbola, and the boundary of the admissible set is a periodic arrangement of congruent overlapping circles. I conjecture that analogous periodic circle arrangements and their associated operators describe Abelian-sandpile scaling limits on general periodic graphs.' 323, 324 thumbnail: "images/flattice.png", 325 thumbnailAnimation: "f-lattice", 326 altText: "Exact parent odometer patterns combining into child pairs along a highlighted branch of the modified Farey tree." 327 }, 328 { 329 id: "hamilton-jacobi-scaling-limits", 330 title: "Hamilton-Jacobi scaling limits of Pareto peeling in 2D", 331 authors: [ 332 { name: "Ahmed Bou-Rabee" }, 333 { name: "Peter S. Morfe", url: "https://sites.google.com/view/pmorfe/home" } 334 ], 335 journal: "Probability Theory and Related Fields", 336 volume: "188", 337 pages: "235-307", 338 year: 2024, 339 links: [ 340 { type: "Journal", url: "https://doi.org/10.1007/s00440-023-01234-4" }, 341 { type: "arXiv", url: "https://arxiv.org/abs/2110.06016" }, 342 { type: "Code", url: "https://github.com/nitromannitol/2d_pareto_peeling" } 343 ], 344 abstract: 'Pareto peeling is an algorithm for sorting large amounts of multivariate data. For random initial data in 2D, we prove that Pareto peeling converges, in the large sample limit, to the viscosity solution of an explicit Hamilton-Jacobi equation. This resolves a <a href="references/caldernotes.pdf">question posed by Jeff Calder</a> and complements earlier work by <a href="https://arxiv.org/abs/1805.08278">Jeff Calder and Charles Smart</a>. While some of our theorems apply in arbitrary dimension, we now think that for d > 2 the process may not always converge.', 345 thumbnail: "images/pareto_peeling.png", 346 thumbnailAnimation: "pareto-peeling", 347 altText: "Successive Pareto fronts peeling a planar point cloud." 348 }, 349 { 350 id: "shape-theorem-exploding-sandpiles", 351 title: "A shape theorem for exploding sandpiles", 352 authors: [ 353 { name: "Ahmed Bou-Rabee" } 354 ], 355 journal: "Annals of Applied Probability", 356 volume: "34(1A)", 357 pages: "714-742", 358 year: 2024, 359 links: [ 360 { type: "Journal", url: "https://doi.org/10.1214/23-AAP1976" }, 361 { type: "arXiv", url: "https://arxiv.org/abs/2102.04422" }, 362 { type: "Notebook", url: "notebooks/exploding_sandpiles.html" } 363 ], 364 abstract: 'Some sandpiles explode: sites topple without stopping. I prove that random, exploding sandpiles have a limit shape. The proof uses ideas from my <a href="https://arxiv.org/abs/2009.05968">dimensional reduction paper</a> to compare the dynamics of an exploding sandpile to a <a href="references/schonmann.pdf">bootstrap percolation</a> process. This project grew out of a question posed by Lionel Levine during <a href="https://math7710.wordpress.com/2020/10/13/october-13-student-research-projects/"> an online topics class he ran during the COVID pandemic.</a>', 365 thumbnail: "images/exploding_pile.png", 366 thumbnailAnimation: "exploding-sandpile", 367 altText: "The toppled region of an exploding random sandpile expanding across the lattice." 368 }, 369 { 370 id: "dynamic-dimensional-reduction", 371 title: "Dynamic dimensional reduction in the Abelian sandpile", 372 authors: [ 373 { name: "Ahmed Bou-Rabee" } 374 ], 375 journal: "Communications in Mathematical Physics", 376 volume: "390", 377 pages: "933-958", 378 year: 2022, 379 links: [ 380 { type: "Journal", url: "https://doi.org/10.1007/s00220-022-04322-z" }, 381 { type: "arXiv", url: "https://arxiv.org/abs/2009.05968" }, 382 { type: "Code", url: "https://github.com/nitromannitol/arbitrarydim_sandpiles" } 383 ], 384abstract: 'Dimensional reduction in sandpiles, the property that slices of higher-dimensional sandpiles correspond to lower-dimensional ones, has been conjectured since <a href="references/LKG.pdf">at least 1990</a>. I prove an exact version of this conjecture. The key insight is that dimensional reduction is closely tied to a discrete parabolic regularity of the sandpile. The proof demonstrates that the "flow" of sandpile dynamics preserves certain derivative bounds, allowing dimensional reduction to occur.', 385 thumbnail: "images/dim_reduction.png", 386 thumbnailAnimation: "dimensional-reduction", 387 altText: "Parallel toppling on a three-dimensional sandpile whose central face has the two-dimensional firing pattern." 388 }, 389 { 390 id: "convergence-random-abelian-sandpile", 391 title: "Convergence of the random Abelian sandpile", 392 authors: [ 393 { name: "Ahmed Bou-Rabee" } 394 ], 395 journal: "Annals of Probability", 396 volume: "49(6)", 397 pages: "3168-3196", 398 year: 2021, 399 links: [ 400 { type: "Journal", url: "https://doi.org/10.1214/21-AOP1528" }, 401 { type: "arXiv", url: "https://arxiv.org/abs/1909.07849" }, 402 { type: "Pictures", url: "https://twitter.com/sandpileofthed1" } 403 ], 404 abstract:"I show that the Abelian sandpile, started with a random initial configuration on any periodic lattice, approximates the solution of a deterministic, fully nonlinear elliptic PDE at large scales. The PDE is implicitly identified through the ergodic theorem, using ideas from non-divergence form stochastic homogenization. Although the exact form of the PDE remains unknown, it is definitively non-universal: it depends on the fine, local structure of both the initial lattice and randomness.", 405 thumbnail: "gallery/plates/random-convergence-hero.png", 406 thumbnailAnimation: "random-sandpile", 407 altText: "Computed stabilization of random 3 or 5 grain heights in disks of increasing radius, shown at the same macroscopic scale." 408 } 409]; 410 411const galleryAnchorByAnimation = { 412 "manhattan-lattice": "manhattan-lattice", 413 "orrw-range": "orrw-range", 414 "parking": "parking", 415 "divisible-percolation": "divisible-percolation", 416 "rotor-walk": "rotor-walk", 417 "einstein-relation": "einstein-relation", 418 "long-range-walk": "long-range-walk", 419 "idla-cylinder": "idla-cylinder", 420 "sandpile-rwrs": "sandpile-rwrs", 421 "algebraic-superdiffusion": "superdiffusion", 422 "sphere-packing": "sphere-packing", 423 "critical-superdiffusion": "superdiffusion", 424 "unique-continuation": "unique-continuation", 425 "percolation-harmonic": "percolation", 426 "mated-crt-idla": "divisible", 427 "harmonic-ball": "divisible", 428 "f-lattice": "f-lattice", 429 "pareto-peeling": "pareto-peeling", 430 "exploding-sandpile": "random-sandpile", 431 "dimensional-reduction": "dimensional-reduction", 432 "random-sandpile": "random-sandpile" 433}; 434 435publications.forEach(publication => { 436 const anchor = galleryAnchorByAnimation[publication.thumbnailAnimation]; 437 if (!anchor || publication.links.some(link => link.type === "Gallery")) return; 438 publication.links.push({ type: "Gallery", url: `gallery/#${anchor}` }); 439});
Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.