Quantum fields and random geometries

2021 Fiscal Year Research-status Report

• Project/Area Number: 21K20340
• Research Institution: Okinawa Institute of Science and Technology Graduate University
• Principal Investigator: DELPORTE Nicolas 沖縄科学技術大学院大学, 重力、量子幾何と場の理論ユニット, ポストドクトラルスカラー (30913199)
• Project Period (FY): 2021-08-30 – 2023-03-31
• Keywords: long-range / fermions

Outline of Annual Research Achievements

- We have understood how the arguments of (Aizenman 1985) for deriving intersections of Brownian random walks (short range propagators) on Galton-Watson trees, adapt to the more general long-range ones, also called Levy flights (cf. the report of our rotation student) and derived critical properties on long-range fields with this random walk point of view. We will explore different questions along those lines with D. Croydon.
- Our original thoughts on fermionic propagators on graphs were naive. Instead, we have obtained a different formulation of a random walk expansion for the Dirac operator on graphs (involving walking on vertices and edges) and we are wondering now, how a two-point function (Green function) could be obtained with a combinatorial expansion through random walks. Then how higher correlators (higher moments) could be computed?
- We have found a nice relation to operator growth bounds on graphs in Quantum information (A. Lucas), and evaluation of Lyapunov exponents through random walks (J. Magan), to determine scrambling. This perspective should provide an answer to compare to, when using instead intersection probabilities of random walks.

Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

We had a rotation student that helped to solve the long-range case and focused for some time on the Dirac fermions. Now that traveling is easier, we will also open collaborations with probabilists in Japan.

Strategy for Future Research Activity

- finish the intersection probabilities with fermionic random walks;
- connections to the "scrambling" on graphs community; does it give a bound precise enough for a generic correlator on a (random graph)? How close to this bound do we get with random walks intersecting?
- longer term goal: obtain renormalization group equations.

Causes of Carryover

We did not spend money from the grant yet, prefering to save it for keeping it for near in time collaboration travels and two workshops at OIST hopefully to come.