Kozma's centrally excited walk converges to a Euclidean ball

A centrally excited random walk on ${\mathbb{Z}}^d$ moves like simple random walk, except that its first step from each site has a drift of fixed size toward the origin. Kozma (2007) conjectured that after $n$ steps the visited set approximates a ball with radius of order $n^{1\mathrm{/}(d+1)}$. We prove this in every dimension $d\ge 2$ and show that the rescaled numbers of visits converge uniformly to a cone, which is the potential generated by the drift on a ball. The same argument shows that a drift opposite to a subgradient of a norm produces the ball of that norm. In the plane, the outer and inner radii of the visited set differ by at most $n^{1\mathrm{/}6}$ times a power of $\log n$, and the exponent $1\mathrm{/}6$ cannot be lowered.