Alexander Dunlap

Bio: Alexander Dunlap is an academic researcher from Stanford University. The author has contributed to research in topics: Burgers' equation & Heat equation. The author has an hindex of 9, co-authored 27 publications receiving 239 citations. Previous affiliations of Alexander Dunlap include Courant Institute of Mathematical Sciences & New York University.

Tightness of Liouville first passage percolation for γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$

Abstract: We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_{t}$ in the bulk, and related star-scale invariant metrics. For $\gamma \in (0,2)$ and $\xi = \frac{\gamma }{d_{\gamma }}$ , where $d_{\gamma }$ is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020), we show that renormalized metrics $(\lambda _{t}^{-1} e^{ \xi p_{t} * h} ds)_{t \in (0,1)}$ are tight with respect to the uniform topology. We also show that subsequential limits are bi-Holder with respect to the Euclidean metric, obtain tail estimates for side-to-side distances, and derive error bounds for the normalizing constants $\lambda _{t}$ .

Fluctuations of the solutions to the KPZ equation in dimensions three and higher

Abstract: We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in $$d\ge 3$$ with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkinson model. This gives an alternative proof, that avoids perturbation expansions, to the results of Magnen and Unterberger (J Stat Phys 171:543–598, 2018).

Tightness of Liouville first passage percolation for $\gamma \in (0,2)$

Abstract: We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_t$ in the bulk, and related star-scale invariant metrics. For $\gamma \in (0,2)$ and $\xi = \frac{\gamma}{d_{\gamma}}$, where $d_{\gamma}$ is the Liouville quantum gravity dimension defined in [Ding-Gwynne18], we show that renormalized metrics $(\lambda_t^{-1} e^{\xi p_t * h} ds)_{t \in (0,1)}$ are tight with respect to the uniform topology. In particular, we show that subsequential limits are bi-Holder with respect to the Euclidean topology, obtain tail estimates for side-to-side distances and derive error bounds for the normalizing constants $\lambda_t$.

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Abstract: The $(d+1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces A deep mathematical understanding of the KPZ equation for $d=1$ has been achieved in recent years, and the case $d\ge 3$ has also seen some progress The most physically relevant case of $d=2$, however, is not very well-understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the $d=2$ case is neither ultraviolet superrenormalizable like the $d=1$ case nor infrared superrenormalizable like the $d\ge 3$ case Moreover, unlike in $d=1$, the Cole-Hopf transform is not directly usable in $d=2$ because solutions to the multiplicative stochastic heat equation are distributions rather than functions In this article we show the existence of subsequential scaling limits as $\varepsilon \to 0$ of Cole-Hopf solutions of the $(2+1)$-dimensional KPZ equation with white noise mollified to spatial scale $\varepsilon$ and nonlinearity multiplied by the vanishing factor $|\log\varepsilon|^{-1/2}$ We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a non-vanishing effect We thus propose our scaling limit as a notion of KPZ evolution in $2+1$ dimensions

Constructing a solution of the $(2+1)$-dimensional KPZ equation

Abstract: The $(d+1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for $d=1$ has been achieved in recent years, and the case $d\ge 3$ has also seen some progress. The most physically relevant case of $d=2$, however, is not very well understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the $d=2$ case is neither ultraviolet superrenormalizable like the $d=1$ case nor infrared superrenormalizable like the $d\ge 3$ case. Moreover, unlike in $d=1$, the Cole–Hopf transform is not directly usable in $d=2$ because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article, we show the existence of subsequential scaling limits as $\varepsilon \to 0$ of Cole–Hopf solutions of the $(2+1)$-dimensional KPZ equation with white noise mollified to spatial scale $\varepsilon $ and nonlinearity multiplied by the vanishing factor $|\log \varepsilon |^{-\frac{1}{2}}$. We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a nonvanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in $2+1$ dimensions.

Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

Abstract: We show that for each $\gamma \in (0,2)$, there is a unique metric (i.e., distance function) associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) $h$, there is a unique random metric $D_h$ associated with the Riemannian metric tensor "$e^{\gamma h} (dx^2 + dy^2)$" on $\mathbb C$ which is characterized by a certain list of axioms: it is locally determined by $h$ and it transforms appropriately when either adding a continuous function to $h$ or applying a conformal automorphism of $\mathbb C$ (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding, Dubedat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $\gamma \in (0,2)$, we conjecture that our metric is the Gromov-Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.