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We compute the high-dimensional limit of the free energy associated with a multi-layer generalized linear model. Under certain technical assumptions, we identify the limit in terms of a variational formula. The approach is to first show…
In this paper, we study energies associated with hypergraphs. More precisely, we obtain results for the incidence and the singless Laplacian energies of uniform hypergraphs. In particular, we obtain bounds for the incidence energy as…
We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models we prove that there is…
We present a technique for calculating free-energy profiles for the nucleation of multicomponent structures that contain as many species as building blocks. We find that a key factor is the topology of the graph describing the connectivity…
We study minimax lower bounds for function estimation problems on large graph when the target function is smoothly varying over the graph. We derive minimax rates in the context of regression and classification problems on graphs that…
We solve explicitly a certain minimization problem for probability measures in one dimension involving an interaction energy that arises in the modelling of aggregation phenomena. We show that in a certain regime minimizers are absolutely…
Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the…
We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral…
We study hierarchical properties of optimal transportation networks with biological background. The networks are obtained as minimizers of an energy functional which involves a metabolic cost term of a power-law form with exponent…
Graph energy is the energy of the matrix representation of the graph, where the energy of a matrix is the sum of singular values of the matrix. Depending on the definition of a matrix, one can contemplate graph energy, Randi\'c energy,…
We study the Laplacian on a metrized graph, and its eigenfunctions.
We consider two different genus expansions of the free energy functions of Hermitian one-matrix models, one using fat graphs, one using ordinary graphs (thin graphs). Some structural results are first proved for the thin version of genus…
In the celebrated work of Friesecke, James and M\"uller '06 the authors derive a hierarchy of models for plates by carefully analyzing the $\Gamma$-convergence of the rescaled nonlinear elastic energy. The key ingredient of their proofs is…
A variety of methods are developed for characterising the free energy landscapes of continuum, Landau-type free energy models. Using morphologies of lipid vesicles and a multistable liquid crystal device as examples, I show that the methods…
We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular…
We compute the continuum limit of the spectra for the XX-model with arbitrary complex boundary fields. In the case of hermitian boundary terms one obtains the partition functions of the free compactified boson field on a cylinder with…
The authors examine graphical properties of eigenfunctions with stadium boundaries associated with avoided crossings of energy levels.
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
We study equilibria in domains with boundaries for a first-order aggregation model that includes social interactions and exogenous forces. Such equilibrium solutions can be connected or disconnected, the latter consisting in a delta…
In this paper we introduce a different kind of graph energy, the minimum 3-covering energy of a graph, and determine the minimum 3-covering energy of complete graphs.