# Chin Lun

Office: D2.08

Email: c.h.lun[at]warwick.ac.uk

I am a postgraduate student at the University of Warwick. I am currently in my fourth year of study in the MASDOC doctoral training centre. Before MASDOC, I graduated with a MMath degree also in the University of Warwick.

## Current PhD Research

My supervisor is Dr Jon Warren. My research interest is in the area of probability, more specifically I am interested in KPZ universality and related topics. I mainly work on the multi-layer extension to the stochastic heat equation introduced by my supervisors. In the paper , the authors defined the following multi-layer process: for $$n=1,2,\ldots$$, $$x$$, $$y\in\mathbb{R}$$

$$Z_n(t,x,y) = p_t(x-y)^n \bigg( 1 + \sum_{k=1}^\infty \int_{\Delta_k(t)} \int_{\mathbb{R}^k} R_k^{(n)}\big( (t_1,x_1), \ldots, (t_k,x_k) \big) \;\prod_{j=1}^k W(\mathrm{d}t_j,\mathrm{d}x_j) \bigg)$$,

where $$R_k^{(n)}$$ is the $$k$$-point correlation function for a collection of $$n$$ non-intersecting Brownian bridges which all start at $$x$$ at time 0 and all end at $$y$$ at time $$t$$, $$p_t(x-y) = (2\pi t)^{-1/2} e^{-(x-y)^2/2t}$$ is the usual heat kernel and $$\Delta_k(t) = \{0<t_1<\cdots<t_k<t\}$$. The integral is a multiple stochastic integral with respect to space-time white noise and the series converges in $$L^2(W)$$. Notice that $$Z_1$$ is the solution to the stochastic heat equation with initial condition $$\delta_x$$ and so $$h = \log Z_1$$ is the Cole-Hopf solution to the KPZ equation with narrow wedge initial condition.

With the Feynman-Kac formula, the Cole-Hopf solution can be interpreted as the continuum analogue of the longest increasing subsequence of a random permutation, length of the first row of a random Young diagram, directed last passage percolation and free energy of a discrete/semi-discrete polymer in random media etc. In each of these discrete models, there is further structure provided either by multiple non-intersecting up-right paths on lattices, multi-layer growth dynamics or Young diagrams constructed from the RSK correspondence.
The work in the above mentioned references have shown that in some cases, utilisation of this additional structure have lead to derivations of exact formulae for the distribution of quantities of interest. The above mentioned discrete models provide examples of what is called integrability or exact solvability. The motivation for introducing the partition functions $$Z_n$$, which are the continuum analogue of the structures mentioned above, is that they should provide insight to the integrable structure in the continuum setting.

The random field $$Z_n(t,x,y)$$ possesses nice regularity properties: for all $$n\geq 1$$, there is a version of $$Z_n$$ that is continuous over $$(0,\infty)\times\mathbb{R}\times\mathbb{R}$$ and almost surely for all $$n\geq 1$$ and $$(t,x,y)$$, $$Z_n(t,x,y)>0$$. Due to the everywhere strict positivity of $$Z_n$$ we can define

$$h_n(t,x) = \log \left(\frac{Z_{n}(t,0,x)}{Z_{n-1}(t,0,x)}\right), \quad n\geq 1, Z_0\equiv 1$$.

The collection $$\{h_n, n\geq 1\}$$ is the analogue in the setting of the KPZ of the multi-layer PNG or its discrete counterpart. It can also be considered as the continuum analogue of the RSK/gRSK correspondence which plays a huge role in the integrability of the above mentioned discrete models. With this RSK interpretation it is natural to ask whether the process

$$(Z_1(t,x,\cdot),\ldots, Z_n(t,x,\cdot) ), t\geq 0$$,

is Markov. This was proven in the case $$n=2$$ in this paper and has more recently been extended to general $$n$$. The proof requires a certain integral formula which suggests a connection with the two-dimensional Toda equation.

It is an ongoing research to see whether for each $$t>0$$, the collection $$\{h_n(t,x), n\geq 1, x\in\mathbb{R}\}$$ determine space-time white noise $$W_t$$ on the time interval $$[0,t]$$ so that the mapping $$W_t \mapsto \{h_n(t,\cdot), n\geq 1\}$$ does represent a correspondence analogous to the usual RSK.