### 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].

**References:**

[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].

**References:**

[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).

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