arXiv subjects

Aukosh Jagannath

Publications and source records attributed to Aukosh Jagannath.

39 records · Page 3Linked to original sources

arXiv2015

A Dynamic Programming Approach to the Parisi Functional

G.Parisi predicted an important variational formula for the thermodynamic limit of the intensive free energy for a class of mean field spin glasses. In this paper, we present an elementary approach to the study of the Parisi functional using stochastic dynamic programing and semi-linear PDE. We give a derivation of important properties of the Parisi PDE avoiding the use of Ruelle Probability Cascades and Cole-Hopf transformations. As an application, we give a simple proof of the strict convexity of the Parisi functional, which was recently proved by Auffinger and Chen in [2].

math.PR↗

arXiv2014

Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

In this paper, we introduce a notion called "Approximate Ultrametricity" which encapsulates the phenomenology of a sequence of random probability measures having supports that behave like ultrametric spaces insofar as they decompose into nested balls. We provide a sufficient condition for a sequence of random probability measures on the unit ball of an infinite dimensional separable Hilbert space to admit such a decomposition, whose elements we call clusters. We also characterize the laws of the measures of the clusters by showing that they converge in law to the weights of a Ruelle Probability Cascade. These results apply to a large class of classical models in mean field spin glasses. We illustrate the notion of approximate ultrametricity by proving two important conjectures regarding mixed p-spin glasses.

math.PR↗

arXiv2014

Solution of the propeller conjecture in $\mathbb{R}^3$

It is shown that every measurable partition ${A_1,..., A_k}$ of $\mathbb{R}^3$ satisfies $$\sum_{i=1}^k||\int_{A_i} xe^{-\frac12||x||_2^2}dx||_2^2\le 9π^2.\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4,...,k\}$ (up to measure zero corrections, orthogonal transformations and renumbering of the sets $\{A_1,...,A_k\}$). This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of $(*)$ is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with $4 \times 4$ centered and spherical hypothesis matrix equals $\frac{2π}{3}$.

cs.CC↗