Supelec Centrale 25: mathematics 2.
In this article you'll find suggested answers for the Maths 2 MP problem set of the Centrale Supelec 2025 competitive exam.
An original and interesting problem, with both linear algebra and analysis, which studies a model of ferromagnetism; we start with a Hamiltonian (the energy of the system) defined by: \[ \begin{alignat*}{1} \{-1,1\}^n\times \mathbb{R}^+ &\to \mathbb{R}\\ (x,h) &\mapsto H(x,h)=-\frac{\beta}{2}\sum_{(i,j)\in\lBrack 1,n\rBrack^2} J_n(i,j)x_ix_j-h\sum_{i=1}^n x_i \end{alignat*} \]
The first term represents the interaction between the particles, the second the effect of the magnetic field.
In the course of the problem, different models are studied, each one characterized by a matrix \(J_n\).
A very long problem, with some really difficult questions.
Correction of the Maths 2 MP problem on May 2nd for the Centrale Supelec 2025 competition:
Find problem statement here: