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We show the convergence of the single sourceless critical random current to a limit identifiable with the nested CLE(3). Our approach is based on viewing the random current as a perturbation of the Ising interface, which is known to converge to CLE(3). Instead of focusing solely on the random current, we provide a general framework for the stability of scaling limits under the perturbation by superimposing an independent Bernoulli percolation., Comment: 40 pages, 7 figures

We study the Cauchy problem of a Hamilton-Jacobi equation with the spatial variable in a closed convex cone. A monotonicity assumption on the nonlinearity allows us to prescribe no condition on the boundary of the cone. We show the well-posedness of the equation in the viscosity sense and prove several properties of the solution: monotonicity, Lipschitzness, and representations by variational formulas., Comment: 35 pages; Fixed an omission of a condition in the comparison principle, Proposition 3.1, which propagates to Corollary 3.2 and Proposition 4.1; More precisely, $\mathsf{F}$ should be in $\Gamma^\nearrow_\cdots$ instead of $\Gamma_\cdots$; Nevertheless, proofs need no change; Also, an appendix is added to contain the alternative versions of these results when $\mathsf{F}$ is not monotone

We give an intrinsic meaning to the Hamilton-Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness. Originally defined on the set of monotone probability measures, these equations can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with an empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existence of solutions. Under additional convexity conditions, we show that the solution can be represented by a version of the Hopf-Lax formula, or the Hopf formula on cones. Previously, two notions of solutions were considered, one defined directly as the Hopf-Lax formula, and another as limits of finite-dimensional approximations. They have been proven to describe the limit free energy in a wide class of mean-field spin glass models. This work shows that these two kinds of solutions are viscosity solutions., Comment: 45 pages; major revision: 1) we directly cited results for finite-dimensional equations from [arXiv:2206.12537] and removed proofs, 2) improved the main result (now independent of the choice of the regularization of the covariance function), and 3) corrected mistakes in handling variational formulas

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 that the limit is a solution to a Hamilton-Jacobi equation whose initial condition is related to the limiting free energy of a model with one fewer layer. Then, we conclude by an iteration., Comment: 43 pages