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.
[JN13] Járai, Antal; Nachmias, Asaf. Electrical resistance of the low
dimensional critical branching random walk.
[K86] Kesten, Harry. Subdiffusive behavior of random walk on a random cluster.
Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425-487.
[KN09] Kozma, Gady; Nachmias, Asaf. The Alexander-Orbach conjecture holds in high
dimensions. Invent. Math. 178 (2009), no. 3, 635-654.
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).
[D06] Dhar, Deepak:
Theoretical studies of self-organized criticality.
Phys. A 369, 29-70 (2006).
[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,
[JW12] Járai, Antal A.; Werning, Nicolás:
Minimal configurations and sandpile measures.
To appear in J. Theor. Probab.
[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.
[MD92] Majumdar, Satya N. and Dhar, Deepak:
Equivalence between the Abelian sandpile model and the
q → 0 limit of the Potts model.
185, 129-145 (1992).
Research Interests |
Publications and Preprints
Department of Mathematical Sciences