Research Papers

My CV; my Google Scholar profile.

D. Blanquicett, J. Gravner, D. Sivakoff, L. Wilson, Two-dimensional supercritical growth dynamics with one-dimensional nucleation (pdf).

J. Gravner, X. Liu, Weakly robust periodic solutions of one-dimensional cellular automata with random rules, Journal of Cellular Automata 17 (2023), 251-280. (pdf).

J. Gravner, B. Kolesnik, Transitive closure in a polluted environment, Annals of Applied Probability 33 (2023), 107-126 (pdf).

J. Gravner, X. Liu, Maximal temporal period of a periodic solution generated by a one-dimensional cellular automaton, Complex Systems 30(2021), 239-272 (pdf).

J. Gravner, X. Liu, One-dimensional cellular automata with random rules: longest temporal period of a periodic solution, Electronic Journal of Probability 27 (2022), 1-23 (pdf).

J. Gravner, X. Liu, Periodic solutions of one-dimensional cellular automata with random rules, Electronic Journal of Combinatorics 28 (2021), #P4.51, 1-19 (pdf).

J. Gravner, D. Sivakoff, Bootstrap percolation on the product of the two-dimensional lattice with a Hamming square, Annals of Applied Probability 30 (2020), 145-174 (pdf).

J. Gravner, K. Johnson, Coupled map lattices as musical instruments, Computer Music Journal 42 (2018), 22-34 (pdf).

M. Damron, J. Gravner, M. Junge, H. Lyu, D. Sivakoff, Parking on transitive unimodular graphs, Annals of Applied Probability 29 (2019), 2089-2113 (arxiv).

J. Gravner, J.E. Paguyo, E. Slivken, Maximal spanning time for neighborhood growth on the Hamming plane, SIAM Journal on Discrete Mathematics 33 (2019), 976-993 (pdf).

J. Gravner, A. E. Holroyd, D. Sivakoff, Polluted bootstrap percolation in three dimensions, Annals of Applied Probability 31 (2021), 218-246 (pdf).

J. Gravner, A. E. Holroyd, Polluted bootstrap percolation with threshold two in all dimensions, Probability Theory and Related Fields 175 (2019), 467-486 (pdf).

J. Gravner, H. Lyu, D. Sivakoff, Limiting behavior of 3-color excitable media on arbitrary graphs, Annals of Applied Probability 28 (2018), 3324-3357 (arxiv).

J. Gravner, D. Sivakoff, E. Slivken, Neighborhood growth dynamics on the Hamming plane, Electronic Journal of Combinatorics 24 (2017) #P4.29, 1-55 (pdf).

J. Gravner, D. Sivakoff, Bootstrap percolation on products of cycles and complete graphs, Electronic Journal of Probability 22 (2017), paper no. 29, 1-20 (arxiv).

J. M. Baetens, J. Gravner, Stability of cellular automata trajectories revisited: branching walks and Lyapunov profiles, Journal of Nonlinear Science 26 (2016), 1329-1367 (arxiv).

J. M. Baetens, J. Gravner, Introducing Lyapunov profiles of cellular automata, Journal of Cellular Automata 13 (2018), 267-286 (pdf). An early version appeared in Proceedings of the 20th International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2014) Himeji, Japan, July 2014," T. Isokawa, K. Imai, N. Matsuin , F. Peper, and H. Umeo, editors, pages 133-140 (arxiv).

J. Gravner, D. Sivakoff, Nucleation scaling in jigsaw percolation, Annals of Applied Probability 27 (2017), 395-438 (arxiv).

J. Gravner, A. E. Holroyd, Percolation and disorder-resistance in cellular automata, Annals of Probability 43 (2015), 1731-1776 (arxiv).

J. Gravner, C. Hoffman, J. Pfeiffer, D. Sivakoff, Bootstrap percolation on the Hamming torus, Annals of Applied Probability 25 (2015), 287-323 (arxiv).

J. Gravner, D. Griffeath, Robust periodic solutions and evolution from seeds in one-dimensional edge cellular automata, Theoretical Computer Science 466 (2012), 64-86 (pdf).

M. Mesterton-Gibbons, S. Gavrilets, J. Gravner, E. Akcay, Models of coalition or alliance formation, Journal of Theoretical Biology 274 (2011), 187-204 (pdf).

J. Gravner, G. Gliner, M. Pelfrey, Replication in one-dimensional cellular automata, Physica D: Nonlinear Phenomena 240 (2011), 1460-1474 (pdf).

J. Gravner, D. Griffeath, The one-dimensional Exactly 1 cellular automaton: replication, periodicity, and chaos from finite seeds, Journal of Statistical Physics 142 (2011), 168-200 (pdf).

J. Gravner, A. E. Holroyd, R. Morris, A sharper threshold for bootstrap percolation in two dimensions, Probability Theory and Related Fields 153 (2012), 1-23 (arxiv).

J. Gravner, D. Griffeath, Asymptotic Densities for Packard Box Rules, Nonlinearity 22 (2009), 1817-1846 (pdf).

J. Gravner, D. Griffeath, Modeling snow-crystal growth: A three-dimensional mesoscopic approach, Physical Review E 79 (2009), 011601, 1-18 (pdf).
Here are some screenshots of popular accounts of this work, from (in order) PhysOrg, Science Daily, CNET, WhyFiles, Chicago Tribune, Grani (Russia), Aftenposten (Norway).

J. Gravner, A. E. Holroyd, Local Bootstrap Percolation, Electronic Journal of Probability 14 (2009), 385-399 (journal).

J. Gravner, Growth phenomena in cellular automata, In "Encyclopedia of Complexity and Systems Science,'' (R. A. Meyers, ed.), Springer, 2009, 15 pages (pdf).

E. M. LaBolle, G. E. Fogg, J. B. Eweis, J. Gravner, D. G. Leaist, Isotopic fractionation by diffusion in groundwater Water Resourses Research 44 (2008), W07405, doi:10.1029/2006WR005264, 1-15 (journal).

J. Gravner, A. E. Holroyd, Slow Convergence in Bootstrap Percolation, Annals of Applied Probability 18 (2008) 909-928. (pdf).

J. Gravner, D. Griffeath, Modeling Snow Crystal Growth II: A mesoscopic lattice map with plausible dynamics, Physica D: Nonlinear Phenomena 237 (2008), 385-404 (pdf).

J. Gravner, S. Gavrilets, D. Pitman, Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities, Journal of Theoretical Biology 248 (2007), 627-645 (arxiv).

J. Gravner, D. Griffeath, Modeling snow crystal growth. I. Rigorous results for Packard's digital snowflakes, Experimental Mathematics 15 (2006), 421-444 (pdf).

J. Gravner, D. Griffeath, Random growth models with polygonal shapes, Annals of Probability 34 (2006), 181-218 (arxiv).

J. Gravner, C. Tracy, H. Widom, Fluctuations in the composite regime of a disordered growth model, Communications in Mathematical Physics 229 (2002), 433-458 (arxiv).

J. Gravner, C. Tracy, H. Widom, A growth model in a random environment, Annals of Probability 30 (2002), 1340-1368 (arxiv).

J. Gravner, C. Tracy, H. Widom, Limit theorems for height fluctuations in a class of discrete space and time growth models, Journal of Statistical Physics 102 (2001), 1085-1132 (arxiv).

S. Gavrilets, R. Acton, J. Gravner, Dynamics of speciation in a metapopulation, Evolution 54 (2000), 1493-1501 (pdf).

J. Gravner, J. Quastel, Internal DLA and the Stefan problem, Annals of Probability 28 (2000), 1528-1562 (journal).

J. Gravner, Growth phenomena in cellular automata, In "New Constructions in Cellular Automata," D. Griffeath and C. Moore, eds., Oxford University Press 2003, pgs. 161-181 (pdf).

J. Gravner, D. Griffeath, Scaling laws for a class of critical cellular automaton growth rules, In "Random Walks," (P. Revesz and B. Toth, eds.) Janos Bolyai Mathematical Society, Budapest 1999, Pgs. 167-186 (pdf).

E. M. LaBolle, J. Quastel, G. E. Fogg, J. Gravner, Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients, Water Resources Research 36 (2000), 651-662 (journal).

T. Bohman, J. Gravner, Random threshold growth dynamics, Random Structures and Algorithms 15 (1999), 93-111 (pdf).

J. Gravner, D. Griffeath, Reverse shapes in first--passage percolation and related growth models, In "Perplexing Problems in Probability. Festshrift in Honor of Harry Kesten," (M. Bramson and R. Durrett, eds.) Birkhauser, 1999. Pgs. 121-142. (pdf).

J. Gravner, Recurrent ring dynamics in two--dimensional excitable cellular automata, Journal of Applied Probability 36 (1999), 492-511 (pdf).

J. Gravner, D. Griffeath, Cellular automaton growth on Z2: theorems, examples and problems, Advances in Applied Mathematics 21 (1998), 241--304 (pdf).

J. Gravner, D. Griffeath, Nucleation parameters in discrete threshold growth dynamics, Experimental Mathematics 6 (1997), 207-220 (pdf).

J. Gravner, D. Griffeath, Multitype threshold voter model and convergence to Poisson-Voronoi tessellation, Annals of Applied Probability 7 (1997), 615-647 (journal).

J. Gravner, E. McDonald, Bootstrap percolation in a polluted environment, Journal of Statistical Physics 87 (1997), 915-927 (pdf).

S. Gavrilets, J. Gravner, Percolation on the fitness hypercube and the evolution of reproductive isolation, Journal of Theoretical Biology 184 (1997), 51-64 (pdf).

J. Gravner, Cellular automata models of ring dynamics, International Journal of Modern Physics C 6 (1996), 863-871 (pdf).

J. Gravner, Percolation times in two-dimensional models for excitable media, Electronic Journal of Probability 1 (1996), paper 12, pages 1-19 (journal).

J. Gravner, D. Griffeath, First passage times for the threshold growth dynamics on Z2, Annals of Probability 24 (1996), 1752-1778 (journal).

J. Gravner, The boundary of iterates in Euclidean growth models, Transactions of the American Mathematical Society 348 (1996), 4549-4559 (journal).

K. L. Verosub, E. G. Puckett, I. Aleinov, D. J. Barsky, J. Gravner, J. Quastel, An Ising model approach to the behavior of Earth's magnetic field: a progress report, Surveys in Geophysics 17 (1996), 225-228 (journal; incredibly, my name was misspelled in the published paper).

R. Fisch, J. Gravner, One-dimensional deterministic Greenberg-Hastings models, Complex Systems 9 (1995), 329--348 (pdf).

R. Fisch, J. Gravner, D. Griffeath, Metastability in the Greenberg-Hastings model, Annals of Applied Probability 3 (1993), 935-967 (journal).

J. Gravner, D. Griffeath Threshold growth dynamics, Transactions of the American Mathematical Society 40 (1993), 837-870 (journal).

R. Fisch, J. Gravner, D. Griffeath, Threshold-range scaling for the excitable cellular automata, Statistic and Computing 1 (1991), 23-39 (pdf).

R. Fisch, J. Gravner, D. Griffeath, Cyclic cellular automata in two dimensions, In "Spatial Stochastic Processes. A Festschrift in honor of Ted Harris." Birkhauser, 1990, pgs. 171-185 (pdf).

J. Gravner A note on elementary operators on the Calkin algebra, Proceedings of the American Mathematical Society 97 (1986), 79-86 (journal).

The research presented here was partially supported by the National Science Foundation and the Republic of Slovenia′s Ministry of Science.