grandes-ecoles 2025 Q19

grandes-ecoles · France · centrale-maths2__official Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.
Verify that $$Z_n(h) = \sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{\beta x_i x_{i+1} + h x_i}$$

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.

Verify that
$$Z_n(h) = \sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{\beta x_i x_{i+1} + h x_i}$$

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