We consider Dyson Brownian motion for classical values of β with deterministic initial data V . We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time t Á 1{N if the density of states of V is bounded above and below down to scales η ! t in a window of size L " ? t. Our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known. Our methodology builds on the homogenization theory developed in [17] which reduces the microscopic problem to a mesoscopic problem. As an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical Dyson Brownian motion.

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We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model. 2010 Mathematics Subject Classification. 35Q55. Key words and phrases. fourth order nonlinear Schrödinger equation; biharmonic nonlinear Schrödinger equation; quasi-invariant measure; normal form method.By iteratively applying the divisor counting argument in B J , . . . , B 2 and then applying (4.16), we have J 3+ sup n∈Z n∈N(π 1 (T J +1 )) nr=n |φ 1 |≥1 n 4s |φ 1 | 2− ψ J ∈Z |ψ J |>(2j+2) 3 j=2,...,J J j=2

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local wellposedness of SSG for any value of a parameter β 2 > 0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: 0 < β 2 < 8π. We also present a triviality result for the unrenormalized SSG.2010 Mathematics Subject Classification. 35L71, 60H15.

We prove that the variance of the passage time from the origin to a point x in first-passage percolation on Z d is sublinear in the distance to x when d ≥ 2, obeying the bound C x / log x , under minimal assumptions on the edge-weight distribution. The proof applies equally to absolutely continuous, discrete and singular continuous distributions and mixtures thereof, and requires only 2+log moments. The main result extends work of Benjamini-Kalai-Schramm [4] and Benaim-Rossignol [6].