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At the mean-field level, on fully connected lattices, several disordered spinmodels have been shown to belong to the universality class of "structuralglasses", with a "random first-order transition" (RFOT) characterized by adiscontinuous jump of the order parameter and no latent heat. However, theirbehavior in finite dimensions is often drastically different, displaying eitherno glassiness at all or a conventional spin-glass transition. We clarify thephysical reasons for this phenomenon and stress the unusual fragility of theRFOT to short-range fluctuations, associated e.g. with the mere existence of afinite number of neighbors. Accordingly, the solution of fully connected modelsis only predictive in very high dimension whereas, despite being alsomean-field in character, the Bethe approximation provides valuable informationon the behavior of finite-dimensional systems. We suggest that before embarkingon a full-blown account of fluctuations on all scales through computersimulation or renormalization-group approach, models for structural glassesshould first be tested for the effect of short-range fluctuations and wediscuss ways to do it. Our results indicate that disordered spin models do notappear to pass the test and are therefore questionable models for investigatingthe glass transition in three dimensions. This also highlights how nontrivialis the first step of deriving an effective theory for the RFOT phenomenologyfrom a rigorous integration over the short-range fluctuations.

On the fragility of the mean-field scenario of structural glasses for finite-dimensional disordered spin models