2022 Fiscal Year Research-status Report
Quantum fields and random geometries
| Project/Area Number |
21K20340
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| Research Institution | Okinawa Institute of Science and Technology Graduate University |
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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)
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| Project Period (FY) |
2021-08-30 – 2024-03-31
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| Keywords | Random walk / Self overlapping curve / Dirac operator / Random Graph |
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| Outline of Annual Research Achievements |
1) We have developed a formalism to rewrite a partition function over two-dimensional metrics on topological disks, of constant curvature, as a random field (with a specific potential), that itself corresponds to self-overlapping curves. Many combinatorial properties of those curves have been studied (average length, area, isoperimetric inequality, gyroscopic radius, winding number, self-intersections). Such a random object provides an intermediate non-Markovian random process between the Brownian motion and the self-avoiding curve, that has tremendous relevance for two dimensional quantum gravity with connections to black holes through the SYK/JT correspondence, for which those curves provide the exact degrees of freedom to comprehend.
2) We have implemented a random walk process generated by an operator corresponding to the square root of a graph Laplacian, giving to the inverse of that operator a combinatorial interpretation (we have applied it to Galton-Watson tree graphs and the Bethe lattice); Through the study of the generating function for the time propagator, we have obtained their heat-kernel, giving a crude estimate of the spectral dimension of the walker (that seem not to differ from the standard random walker, ie 4/3 and 3 respectively) ; In order to exploit general relations obtained for generating functions of pointed graphs (especially trees and planar maps), we understood that it is their asymptotics (close to their singularities) that matter in the computation of n-point correlation functions of fields on the graphs.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
- Finalising the statistical results of 1) (to extend the regime all the way to large enough variance for self-overlaps to be typical);
- For 2), comparing our results to existing literature on quantum random walks; in particular, looking if this random process admits a probabilistic interpretation like the Brownian random walk and if we can define a continuum limit to it, different from the universality class of the Brownian motion (although the spectral dimension results would point to the same class).
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| Strategy for Future Research Activity |
- 1) Implementing a fast algorithm to count distinct disks from a given self-overlapping curve; computation of partition function;
- 2) Fine properties of the fermionic random walks: intersections probabilities, mean first passage time and expected number of sites visited; Does this random walk have any relevance for studying fermionic systems? (studying operator growth or Anderson delocalization transition); link to p-adic field theory.
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| Causes of Carryover |
Because of the Japanese frontiers being closed until the last fiscal year, it was difficult for our collaborators to visit us at OIST and more importantly, the two conferences that we organised at OIST to present our results to the different interested communities were postponed to 2023 (respectively April and August).
We used part of the fund to invite two collaborators to work at OIST end of March 2023-mid April 2023, during the April program Invitation to Recursion, Resurgence and Combinatorics. Part of the fund (1.5 Million Yen) served to supplement OIST funding during this program (4-14 April 2023), for flights of invited speakers, meals, etc.
The remaining fund (1 Million Yen) will serve for the second workshop in August (31July-4August 2023), for supporting flights of invited speakers, meals and transportation.
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