Markov processes of infinitely many nonintersecting random walks
Let \(X(t)=(X_1(t),\dotsc, X_N(t))\in {\mathbb Z}^N\) be \(N\geq 1\) independent rate one Poisson processes started at \(X(0)=(0, 1, \dotsc, N-1)\) and conditioned to finish at \(X(T)=Y\) while taking mutually distinct values for all \(0\leq t\leq T\). Thus, \(\{X_i(t)\}_{i\geq 1}\) form \(N\) nonintersecting paths in \({\mathbb Z}\times [0, T]\). A result of [\textit{W. König} et al., Electron. J. Probab. 7, Paper No. 5 (2002; Zbl 1007.60075)] says that as \(T\to \infty\) with \(Y\) being asymptotically linear, \(Y\sim T\xi\) with a collection of asymptotic speeds \(\xi=(\xi_1\leq \dotsb\leq \xi_N) \in{\mathbb R}_{>0}^N\), the process \((X(t), 0\leq t\leq T)\) has a limit \((\chi(t), t\geq 0)\), which is a homogeneous Markov process on \({\mathbb Z}^N\) with initial condition \(X(0)\) and transition probabilities over time interval \(t\) given by \(\mathrm{const} {{\Xi_N(y)}\over{\Xi_N(x)}}\det_{i,j=1,\dotsc, N}\left[{{t^{y_i - x_j}}\over {(y_i - x_j)!}}\right]\), where \(\Xi_N(u)=\det\left[\xi_i^{u_j}\right]_{i, j=1}^N\). The process \(\Xi(t)\) is the Doob \(h\)-transform of \(N\) independent Poisson processes with respect to the harmonic function \(h=\Xi_N\). In the case of equal speeds, the distribution of \(X(t)\) is known as the Charlier orthogonal polynomial ensemble, which is the basic discrete probabilistic model of random matrix type. The procedure of passing to a limit by increasing the number of particles and scaling the space appropriately in models of random matrix type is very well developed. In many cases, such a limit can also be successfully performed for joint distributions at finitely many times if the original model undergoes a Markov dynamics. The most common approach is to control the limiting behavior of local correlation functions. It is much less clear what happens to the Markovian structure of the dynamics under such limit transitions. One goal of the paper is to prove that, if \(\xi_j\)'s form a geometric progression with ratio \(q^{-1}>1\), the limit \(\Upsilon(t)\) of \(\chi(t)\) as \(N\to \infty\) can be viewed as a Feller Markov process on all point configurations in \({\mathbb Z}_{\geq 0}\) started from the densely packed initial condition. The authors show that the dynamical correlation functions of \(\Upsilon(t)\) started from the packed initial condition are determinantal with an explicit correlation kernel, and they are the limits of the corresponding correlation functions for \(\chi(t)\). In the paper [Adv. Math. 229, No. 1, 201--266 (2012; Zbl 1235.22028)], the second author proved that infinite point configurations in \({\mathbb Z}\) with finitely many particles to the left of the origin can be identified with ergodic \(q\)-Gibbs measures on infinite Gelfand-Tsetlin schemes. Using this fact, the authors show that the Markov processes \(\chi(t)\) for different \(N\) are consistent with respect to natural projections from the \(N\)-particle space to the \((N-1)\)-particle one. The projections are uniquely determined by the \(q\)-Gibbs property. Together with certain (nontrivial) estimates, this leads to the existence of the limiting Feller Markov process.
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