Extremal properties of the random walk local time (2502.09853v1)

Abstract: These are expanded lecture notes for a minicourse taught at the "School on disordered media" at the Alfred Renyi institute in Budapest, January 2025.

• The paper rigorously connects 2D random walk local time with the Gaussian Free Field using potential theory and precise Green function estimates.
• The paper analyzes scaling limits and the distribution of avoided points, uncovering a fractal pattern in extreme local time behaviors.
• The paper employs technical lemmas, such as Poissonian excursions and harmonic decompositions, to enhance the understanding of cover time and extremal statistics.

Extremal Properties of the Random Walk Local Time

The document at hand, authored by Marek Biskup, is a comprehensive set of lecture notes addressing extremal properties of the local time of random walks, presented at a minicourse during the "School on Disordered Phenomena." The focus of the paper is on elucidating the complex behaviors and properties associated with the scaling limits of points avoided by a two-dimensional simple random walk. The document is both an introduction to and an exposition on the interplay between random walk local time and logarithmically correlated processes, such as the Gaussian Free Field (GFF).

Random Walk Local Time

The introductory section defines local time for continuous-time Markov chains on finite state spaces, detailing how local time manifests as a two-parameter stochastic process. The main question of interest in this exploration is the behavior of local time as time progresses, particularly focusing on maximum and minimum local time values, and how these relate to the notion of cover time. Cover time is defined as the first time at which every vertex in the state space has been visited and acts as a critical time scale for understanding the dispersion and regularity of the random walk path.

Scaling Limits and Avoided Points

Central to the document is the discussion of scaling limits relating to points not yet visited by the random walk—referred to as 'avoided points.' Special attention is given to the structure and distribution of these points as they evolve over time within domains governed by random boundaries, such as those found in scaled-up lattice models. For the two-dimensional case, in particular, the document explores how avoided points form a random fractal pattern and aligns this understanding with established theorems concerning the GFF.

Connection to the Gaussian Free Field

The work intricately links local time results to the GFF, a central theme in the analysis of statistical mechanics and probability. By employing Green function asymptotics and properties of the Poisson kernel, Biskup establishes a rigorous connection between local time processes and a Gaussian process termed the Discrete Gaussian Free Field (DGFF). The document explains how the local time of a Markov chain and the DGFF are intertwined through potential theory and harmonic measures, particularly by establishing equivalences and bounds on local time via the GFF.

Main Results and Implications

Key results include an asymptotic characterization of the DGFF’s extremal properties, utilizing concepts such as conformal radius and harmonic measure, and detailed exploration of the maximum and thick points of this field. These results have broader implications for understanding the scaling behaviors of random walks in two-dimensional settings, offering a glimpse into the universality of these behaviors beyond specific lattice configurations or boundary conditions.

Mathematical Rigor and Technical Details

The notes are replete with technical lemmas and proofs, including discussions of Poissonian excursions, Green function representation, and intricate decomposition using harmonic extensions and Ray-Knight theorems. These mathematical tools are critical for developing the theoretical underpinnings needed to fully understand the extremal behaviors of local time. The rigorous approach ensures that complex theoretical constructs are mathematically valid and applicable to broad scenarios in disordered systems.

Future Developments and Open Questions

The document suggests future research avenues, such as adjusting local time frameworks to broader classes of Markov processes and extending the complexity of random environments tackled. Another critical aspect for potential exploration is the application of these results to more generalized settings, such as non-simply connected domains or free boundary conditions, which remain complex yet intriguing extensions of the work presented.

In conclusion, Marek Biskup's lecture notes present a well-structured, insightful exploration of extremal properties of random walk local time, utilizing a blend of stochastic processes, potential theory, and Gaussian processes. The document not only elucidates the intrinsic nature of random walks and their extremal points but also serves as a critical groundwork for future inquiries into other dimensionally constrained and complex stochastic phenomena.