We construct a strongly local symmetric Dirichlet form on the configuration space \(\varvec{\Upsilon }\) whose symmetrising (thus also invariant) measure is \(\textsf{sine}_\beta \) , which is the law of the sine \(\beta \) ensemble for every \(\beta >0\) . For every \(\beta >0\) , this Dirichlet form satisfies the Bakry–Émery gradient estimate \(\textsf {BE} (K, \infty )\) with \(K=0\) . This implies various functional inequalities, including the local Poincaré inequality, the local log–Sobolev inequality and the local hyper-contractivity. We then introduce an \(L^2\) -transportation-type extended distance \(\bar{\textsf {d} }_{\varvec{\Upsilon }}\) on \(\varvec{\Upsilon }\) , and prove the dimension-free Harnack inequality and several Lipschitz regularisation estimates of the \(L^2\) -semigroup associated with the Dirichlet form in terms of \(\bar{\textsf {d} }_{\varvec{\Upsilon }}\) . As a result of \(\textsf {BE} (0,\infty )\) , we obtain that the dual semigroup on the space of probability measures over \(\varvec{\Upsilon }\) , endowed with a Benamou–Brenier-like extended distance \(\textsf {W} _{\mathcal {E}}\) , satisfies the evolutional variation inequality with respect to the Bolzmann–Shannon entropy \(\textsf {Ent} _{\textsf{sine}_\beta }\) associated with \(\textsf{sine}_\beta \) . Furthermore, the dual semigroup is characterised as the unique \(\textsf {W} _{\mathcal {E}}\) -gradient flow in the space of probability measures with respect to \(\textsf {Ent} _{\textsf{sine}_\beta }\) . These results provide quantitative estimates of the transition semigroup of the unlabelled infinite Dyson Brownian motion (DBM) with the inverse temperature \(\beta \) , and give a new perspective regarding the DBM as the \(\textsf {W} _{\mathcal {E}}\) -gradient flow of the Bolzmann–Shannon entropy. Finally, we provide a sufficient condition for \(\textsf {BE} (K, \infty )\) beyond \(\textsf{sine}_\beta \) and apply it to the infinite particle diffusion whose symmetrising measure is the law of the 1-dimensional \((\beta ,s)\) -circular Riesz gas with \(\beta >0\) and \(0<s<1\) .
