[math] \newcommand{\mathds}{\mathbb}[/math]

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Compute the laws of the truncated characters for the main tori

[[math]] \xymatrix@R=50pt@C=50pt{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] } [[/math]]

and then work out the asymptotics, with [math]N\to\infty[/math].