Antal A. Járai: Research interests

General areas of interests

Mathematical models of statistical physics, critical phenomena, self-organized criticality, mean-field behaviour

Specific areas of interest
PhD projects available

PhD projects are available in two different areas.

1) Random walk on random fractals. Random walks on fractal structures produced by random processes are a source of fascinating open problems. The pioneering work of Kesten [K86] showed that random walk on a critical percolation cluster moves significantly slower than diffusive. Several critical exponents for the walk have been established rigorously in the context of high-dimensional oriented and unoriented percolation [BJKS08,KN09]. Recent work has extended this to situations where the exponents are not mean-field [JN13].
[BJKS08] Barlow, Martin T.; Járai, Antal A.; Kumagai, Takashi; Slade, Gordon. Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Comm. Math. Phys. 278 (2008), no. 2, 385-431. 2372764
[JN13] Járai, Antal; Nachmias, Asaf. Electrical resistance of the low dimensional critical branching random walk. Preprint (2013).
[K86] Kesten, Harry. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425-487. 0871905
[KN09] Kozma, Gady; Nachmias, Asaf. The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178 (2009), no. 3, 635-654. 2551766

2) Abelian sandpile model. The Abelian sandpile model has been discovered a number of times independently in different contexts: statistical physics, combinatorics, algebraic geometry and mathematics education. It is a source of difficult mathematical problems. For a brief introduction to the model, see [LP10]. See also [D06] for an extensive survey with motivation from statistical physics. Due to a discovery of [MD92], the Abelian sandpile model has a close connections to uniform spanning trees [LyPe]. This connection has been fruitful in analyzing questions about the sandpile model [AJ04,JR08,JW12].
[AJ04] Athreya, Siva R. and Járai, Antal A.: Infinite volume limit for the stationary distribution of Abelian sandpile models. Comm. Math. Phys. 249 197-213 (2004). 2077255
[D06] Dhar, Deepak: Theoretical studies of self-organized criticality. Phys. A 369, 29-70 (2006). 2246566
[JR08] Járai, Antal A.; Redig, Frank. Infinite volume limit of the abelian sandpile model in dimensions d >= 3. Probab. Theory Related Fields 141 (2008), no. 1-2, 181-212. 2372969
[JW12] Járai, Antal A.; Werning, Nicolás: Minimal configurations and sandpile measures. To appear in J. Theor. Probab. Preprint.
[LP10] Levine, Lionel; Propp, James: What is ... a sandpile? Notices Amer. Math. Soc. 57, no. 8, 976-979 (2010).
[LyPe] Lyons, Russell with Peres, Yuval: Probability on trees and networks. Book in preparation.
[MD92] Majumdar, Satya N. and Dhar, Deepak: Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model. Phys. A 185, 129-145 (1992).

Research Interests | Publications and Preprints
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