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We present a new algorithm to sample the constrained eigenvalues of theinitial shear field associated with Gaussian statistics, called the `peak/dipexcursion-set-based' algorithm, at positions which correspond to peaks or dipsof the correlated density field. The computational procedure is based on a newformula which extends Doroshkevich's unconditional distribution for theeigenvalues of the linear tidal field, to account for the fact that halos andvoids may correspond to maxima or minima of the density field. The ability todifferentiate between random positions and special points in space around whichhalos or voids may form (peaks/dips), encoded in the new formula and reflectedin the algorithm, naturally leads to a straightforward implementation of anexcursion set model for peaks and dips in Gaussian random fields - one of thekey advantages of this sampling procedure. In addition, it offers novelinsights into the statistical description of the cosmic web. As a firstphysical application, we show how the standard distributions of shearellipticity and prolateness in triaxial models of structure formation aremodified by the constraint. In particular, we provide a new expression for theconditional distribution of shape parameters given the density peak constraint,which generalizes some previous literature work. The formula has importantimplications for the modeling of non-spherical dark matter halo shapes, inrelation to their initial shape distribution. We also test and confirm ourtheoretical predictions for the individual distributions of eigenvaluessubjected to the extremum constraint, along with other directly relatedconditional probabilities. Finally, we indicate how the proposed samplingprocedure naturally integrates into the standard excursion set model,potentially solving some of its well-known problems, and into the ellipsoidalcollapse framework. (abridged)

Peaks and dips in Gaussian random fields: a new algorithm for the shear eigenvalues, and the excursion set theory