  I present here a pedagogical introduction to the works by Rashel Tublin and
Yan V. Fyodorov on random linear systems with quadratic constraints, using
tools from Random Matrix Theory and replicas. These notes illustrate and
complement the material presented at the Summer School organised within the
Puglia Summer Trimester 2023 in Bari (Italy). Consider a system of $M$ linear
equations in $N$ unknowns, $\sum_{j=1}^N A_{kj}x_j=b_k$ for $k=1,\ldots,M$,
subject to the constraint that the solutions live on the $N$-sphere,
$x_1^2+\ldots + x_N^2=N$. Assume that both the coefficients $A_{ij}$ and the
parameters $b_i$ be independent Gaussian random variables with zero mean. Using
two different approaches -- based on Random Matrix Theory and on a replica
calculation -- it is possible to compute whether a large linear system subject
to a quadratic constraint is typically solvable or not, as a function of the
ratio $\alpha=M/N$ and the variance $\sigma^2$ of the $b_i$'s. This is done by
defining a quadratic loss function $H({\bf
x})=\frac{1}{2}\sum_{k=1}^M\left[\sum_{j=1}^NA_{kj} x_j-b_k\right]^2$ and
computing the statistics of its minimal value on the sphere,
$E_{min}=\min_{||\bf x||^2=N}H({\bf x})$, which is zero if the system is
compatible, and larger than zero if it is incompatible. One finds that there
exists a compatibility threshold $0<\alpha_c<1$, such that systems with
$\alpha>\alpha_c$ are typically incompatible. This means that even weakly
under-complete linear systems could become typically incompatible if forced to
additionally obey a quadratic constraint.
