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I present here a pedagogical introduction to the works by Rashel Tublin andYan V. Fyodorov on random linear systems with quadratic constraints, usingtools from Random Matrix Theory and replicas. These notes illustrate andcomplement the material presented at the Summer School organised within thePuglia Summer Trimester 2023 in Bari (Italy). Consider a system of $M$ linearequations 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 theparameters $b_i$ be independent Gaussian random variables with zero mean. Usingtwo different approaches -- based on Random Matrix Theory and on a replicacalculation -- it is possible to compute whether a large linear system subjectto a quadratic constraint is typically solvable or not, as a function of theratio $\alpha=M/N$ and the variance $\sigma^2$ of the $b_i$'s. This is done bydefining a quadratic loss function $H({\bfx})=\frac{1}{2}\sum_{k=1}^M\left[\sum_{j=1}^NA_{kj} x_j-b_k\right]^2$ andcomputing 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 iscompatible, and larger than zero if it is incompatible. One finds that thereexists a compatibility threshold $0<\alpha_c<1$, such that systems with$\alpha>\alpha_c$ are typically incompatible. This means that even weaklyunder-complete linear systems could become typically incompatible if forced toadditionally obey a quadratic constraint.

Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back