Complete confinement of long-range Riesz gases by a planar wall

This paper determines when a quadratically confined Riesz gas in dimension d ≥ 2 becomes completely supported on a planar hard wall. For the interaction |x−y|^(−s)/s, with logarithmic continuation at s = 0, and −2 < s ≤ d−3, the complete-confinement threshold is positive and finite and is given by an explicit maximization along the normal axis. The wall measure is a Beta law on a ball for d−5 < s ≤ d−3 and a uniform spherical measure for s ≤ d−5. The proof establishes the full half-space variational inequality by a positive convolution comparison for Beta laws and a concavity argument along a moving transverse minimum for spherical laws. Together with known results for d−3 < s < d, this gives complete confinement at some wall position precisely when s ≤ d−2. Further results include separation of critical off-wall contact from the wall, stability above threshold, continuous differentiability of the minimum energy, one-sided energy-deficit bounds below threshold, and a two-term high-dimensional expansion at the spherical endpoint. For s < 0, the deficit bounds rule out twice differentiability of the energy at the threshold.