scholarly journals The Parisi Formula has a Unique Minimizer

2014 ◽  
Vol 335 (3) ◽  
pp. 1429-1444 ◽  
Author(s):  
Antonio Auffinger ◽  
Wei-Kuo Chen
Keyword(s):  
Unique Minimizer ◽  
Parisi Formula

scholarly journals The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer

2021 ◽  
Vol 74 (3) ◽  
pp. 615-675
Author(s):  
Matthias Erbar ◽  
Martin Huesmann ◽  
Thomas Leblé
Keyword(s):  
Free Energy ◽  
Unique Minimizer ◽  
One Dimensional ◽  
The One

scholarly journals Random assignment and shortest path problems

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Johan Wästlund
Keyword(s):  
Limit Theorem ◽  
Shortest Path ◽  
Complete Graph ◽  
Assignment Problem ◽  
Shortest Path Problem ◽  
Random Assignment ◽  
Shortest Path Problems ◽  
Parisi Formula ◽  
International Audience ◽  
Random Assignment Problem

International audience We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.

scholarly journals FREE ENERGY IN THE GENERALIZED SHERRINGTON–KIRKPATRICK MEAN FIELD MODEL

2005 ◽  
Vol 17 (07) ◽  
pp. 793-857 ◽  
Author(s):  
DMITRY PANCHENKO
Keyword(s):  
Free Energy ◽  
Real Line ◽  
Prior Distribution ◽  
Field Model ◽  
Mean Field ◽  
Rigorous Proof ◽  
Bounded Subset ◽  
Mean Field Model ◽  
The Real ◽  
Parisi Formula

In [11], Talagrand gave a rigorous proof of the Parisi formula in the classical Sherrington–Kirkpatrick (SK) model. In this paper, we build upon the methodology developed in [11] and extend Talagrand's result to the class of SK type models in which the spins have arbitrary prior distribution on a bounded subset of the real line.

Diagonal non-semicontinuous variational problems

2018 ◽  
Vol 24 (4) ◽  
pp. 1333-1343
Author(s):  
Sandro Zagatti
Keyword(s):  
Sobolev Spaces ◽  
Hessian Matrix ◽  
Variational Problems ◽  
Convex Envelope ◽  
Minimum Problem ◽  
Unique Minimizer ◽  
Special Case ◽  
Minimum Points

We study the minimum problem for non sequentially weakly lower semicontinuos functionals of the form F(u)=∫If(x,u(x),u′(x))dx, defined on Sobolev spaces, where the integrand f:I×ℝm×ℝm→ℝ is assumed to be non convex in the last variable. Denoting by f̅ the lower convex envelope of f with respect to the last variable, we prove the existence of minimum points of F assuming that the application p↦f̅(⋅,p,⋅) is separately monotone with respect to each component pi of the vector p and that the Hessian matrix of the application ξ↦f̅(⋅,⋅,ξ) is diagonal. In the special case of functionals of sum type represented by integrands of the form f(x, p, ξ) = g(x, ξ) + h(x, p), we assume that the separate monotonicity of the map p↦h(⋅, p) holds true in a neighbourhood of the (unique) minimizer of the relaxed functional and not necessarily on its whole domain.

scholarly journals The Legendre Structure of the Parisi Formula

2016 ◽  
Vol 348 (3) ◽  
pp. 751-770 ◽  
Author(s):  
Antonio Auffinger ◽  
Wei-Kuo Chen
Keyword(s):  
Parisi Formula

scholarly journals The Dispersion Tensor and Its Unique Minimizer in Hashin–Shtrikman Micro-structures

2018 ◽  
Vol 230 (2) ◽  
pp. 665-700 ◽  
Author(s):  
Loredana Bălilescu ◽  
Carlos Conca ◽  
Tuhin Ghosh ◽  
Jorge San Martín ◽  
Muthusamy Vanninathan
Keyword(s):  
Unique Minimizer ◽  
Dispersion Tensor ◽  
Micro Structures

Discrete Möbius energy

2014 ◽  
Vol 23 (09) ◽  
pp. 1450045 ◽  
Author(s):  
Sebastian Scholtes
Keyword(s):  
Discrete Version ◽  
Unique Minimizer ◽  
Discrete Curves

We investigate a discrete version of the Möbius energy, that is of geometric interest in its own right and is defined on equilateral polygons with n segments. We show that the Γ-limit regarding Lq or W1,q convergence, q ∈ [1, ∞] of these energies as n → ∞ is the smooth Möbius energy. This result directly implies the convergence of almost minimizers of the discrete energies to minimizers of the smooth energy if we can guarantee that the limit of the discrete curves belongs to the same knot class. Additionally, we show that the unique minimizer amongst all polygons is the regular n-gon. Moreover, discrete overall minimizers converge to the round circle.

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