RANDOM WALK IN RANDOM ENVIRONMENT

GRADUATE COURSE FOR PhD STUDENTS AND POSTDOCS AT BUDAPEST (BME & ELTE & RI)

2026 SPRING SEMESTER 

BÁLINT TÓTH (Rényi Institute Budapest)

WHEN?

LECTURES  (most likely) HELD ON Thursday 2-4 pm ...... schedule to be announced soon

FIRST LECTURE: on the week 9-13 Feb 2026  ...... schedule to be announced soon

LAST LECTURE: ...... schedule to be announced soon

WHERE?

THE LECTURES WILL BE (most likely) HELD AT BME-H-306  ......  to be announced soon

MESSAGES TO STUDENTS ATTENDING THE COURSE:

8 JANUARY 2026:

• I KINDLY ASK THOSE INTERESTED IN TAKING THIS COURSE TO SEND ME AN INFORMAL EMAIL MESSAGE ABOUT THEIR INTENTION. 

• THE COURSE WILL BE HELD ONLY IF A MINIMUM NUMBER OF 10 ATTENDANTS "REGISTER" (INFORMALLY)

SYLLABUS:

• Survey of main results about simple symmetric random walk on Z^d: Recurrence vs. Transience, Póllya's theorem; Laws of Large Numbers; Central Limit Theorem; Invariance principle

• Random walk in Random Environment (RWRE): motivating examples

•  RWRE in one dimension (on Z^1): Recurrence vs transience; Limit theorems in the recurrent case; Stable limit laws in the sub-ballistically transient cases; some other examples and results

• The reversible case: RW among random conductances, The environment process: reversibility. Martingale approximation: the theorem of Kipnis and Varadhan. Annealed CLT. Quenched vs. annealed. 

• RW in divergence-free random environment: motivating examples and new aspects of the problem. The environment process: stationarity but no reversibility. Diffusive bounds. 

• Non-reversible Kipnis-Varadhan theory and its application to RW in div-free RE: annealed CLT. 

• Harmonic coordinates and quenched CLT. 

• Superdiffusivity in low dimensions. 

• Selection of other problems -- if time permits . 

PREREQUISITS: 

• • • Fluency in probability theory (including conditional expectation (a la Kolmogorov), Markov chains and Markov processes, laws of large numbers, 0-1 laws, ergodicity, central limit theorems)

    • Fluency in analysis and functional analysis (including  basics of Fourier transform, L^p spaces, analysis in Hilbert space, spectral theorem for self-adjoint operators)

However, those who miss some of these elements and therefore hesitate about attending should not shy away. The necessary background knowledge can be picked up on the way presuming enough basic mathematical interest and open mind. 

LECTURE NOTES:

1. Warming up: SSRW and motivating RWRE [last modified 2023-10-18]

2. One-dimension -- strong laws and limit theorems [last modified 2023-10-22]

3. Random walk among random conductances (incl. Kipnis Varadhan Thm - reversible setting) [last modified 2023-11-06]

4. Random walk in divergence-free random environment [last modified 2023-11-14]

5. Martingale approximation and Kipnis-Varadhan theory -- non-reversible setting [last modified 2023-11-13]

6. Quenched CLT and Outlook [last modified 2023-12-08]

7. more to come soon