High-dimensional Gaussian fields with isotropic increments seen through spin glasses

We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle-point variational representation in terms of a Parisi-type functional for the free energy in the infinite-dimensional limit. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.Comment: 13 page

The Aizenman-Sims-Starr and Guerra's schemes for the SK model with multidimensional spins

We prove upper and lower bounds on the free energy in the Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in terms of the variational inequalities based on the corresponding Parisi functional. We employ the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the generalised random energy model-inspired processes and Ruelle's probability cascades. For this purpose an abstract quenched large deviations principle of the Gaertner-Ellis type is obtained. Using the properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent, we derive Talagrand's representation of the Guerra remainder term for our model. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of the non-linear partial differential equations. Solving a problem posed by Talagrand, we show the strict convexity of the local Parisi functional. We prove the Parisi formula for the local free energy in the case of the multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of the a priori estimates.Comment: 60 pages, published versio

Fluctuations of the partition function in the GREM with external field

We study Derrida's generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by strength of external field and by parameters of the GREM.Comment: 24 page

The grapheme-valued Wright-Fisher diffusion with mutation

In [Athreya, den Hollander, R\"ollin; 2021, arXiv:1908.06241] models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterised as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set, an adjacency matrix and a sampling measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Poisson-Dirichlet distribution. In a companion paper [Greven, den Hollander, Klimovsky, Winter; 2023], we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.Comment: 25 page

Complex Random Energy Model: Zeros and Fluctuations

The partition function of the random energy model at inverse temperature $\beta$ is a sum of random exponentials $Z_N(\beta)=\sum_{k=1}^N \exp(\beta \sqrt{n} X_k)$, where $X_1,X_2,...$ are independent real standard normal random variables (= random energies), and $n=\log N$. We study the large $N$ limit of the partition function viewed as an analytic function of the complex variable $\beta$. We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $\beta$, both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.Comment: 31 pages, 1 figur