The "shape" of an abstract mathematical group can be discovered by watching a virtual particle wander through it at random.
April 14, 2026
Original Paper
Spectral Dehn functions and a characterisation of word-hyperbolicity
arXiv · 2604.09014
The Takeaway
It links the geometry of groups to the physics of "random walks," using energy gaps to identify shapes. This allows mathematicians to tell the difference between complex geometric structures that previously appeared identical.
From the abstract
We introduce a \emph{spectral Dehn function} \[ \Lambda_{\mathcal{P}}(n):=\inf \lambda_1(\Delta), \] where $\lambda_1(\Delta)$ is the first Dirichlet eigenvalue of the random-walk Laplacian on a van Kampen diagram $\Delta$, and the infimum runs over area-minimising diagrams with boundary length at most $n$. We prove a spectral-isoperimetric inequality relating $\Lambda_{\mathcal{P}}$ to the Dehn function, and show that its degree-free face-dual variant $\Lambda^\ast_{\mathcal P}$ characterises w