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Probability Theory and Random Point Processes Lab
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A lab that explores random arrangements of points through probability theory.
AboutAbout the lab
Reading random arrangements of points through probability theory.
Professor Tomoyuki Shirai studies probability theory, especially point processes, which describe random arrangements of points.
Building on the properties of eigenvalues of random matrices, the lab constructs random point fields called determinantal point processes and explores their generalizations.
The lab is also interested in the relationships among random walks on graphs, the spectrum of the Laplacian, and the geometric properties of graphs, and has recently begun research on random topology.
Key terms
- Determinantal point process
- A random arrangement of points (a random point field) constructed by abstracting the fermionic properties of the eigenvalues of random matrices
- Random matrix
- A matrix whose entries are determined probabilistically. The way its eigenvalues are arranged serves as a guide for the research
- Random walk
- A process that moves probabilistically from vertex to vertex on a graph
ResearchResearch themes
Using properties observed in the eigenvalues of random matrices as a guide, the lab constructs determinantal point processes, which describe random collections of points, and explores their generalizations. It also works on random walks on graphs, the spectrum of the Laplacian, and random topology.
*Illustrative imageConstruction and Generalization of Determinantal Point Processes
By abstracting the fermionic properties of the eigenvalues of random matrices, the lab constructed determinantal point processes (fermionic random fields). It has become clear that objects such as representations of the infinite symmetric group and the distribution of zeros of random power series can be described by these point processes, and the lab is now exploring generalizations in other directions.
*Illustrative imageRandom Walks on Graphs and Geometry
Random walks on graphs, the spectrum of the Laplacian, and the geometric properties of graphs are closely related to one another, and the lab works to clarify these relationships. It has also recently been conducting research on random topology.
MembersFaculty
- ProfessorTomoyuki Shirai
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “A boolean model” Tensorproduct / CC BY-SA 4.0 (cropped)
- Construction and Generalization of Determinantal Point Processes: “Random and Nonrandom Points Artifact” Caitlin Jo Ramsey / CC BY-SA 3.0 (cropped)
- Random Walks on Graphs and Geometry: “Dodecahedron-snapshot 2” Randdv259 / CC0 (cropped)
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