Abstract
We consider the geodesic of the directed last passage percolation with iid exponential weights. We find the explicit one-point distribution of the geodesic location joint with the last passage times, and its limit as the parameters go to infinity under the KPZ scaling.
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Notes
In fact, we can view the integrand of (1.8) as a function of \(V^{(1)}\) and \(V^{(2)}\), which equals to the product of the following three terms: \(\Delta (V^{(1)})\Delta (V^{(2)})\), a Cauchy-type factor \({\mathcal {C}}(V^{(1)};V^{(2)})=\Delta (V^{(1)})\Delta (V^{(2)}) /\Delta (V^{(1)};V^{(2)})\) (see the definition in (2.48)), and some function which is meromorphic for each \(v_{i_\ell }^{(\ell )}\) with a possible pole at 0 but the degree of this pole is at most \(\max \{n,N-n+1\}\). Note that expanding the first term \(\Delta (V^{(1)})\Delta (V^{(2)})\) gives a sum of terms \(\prod _{1\le \ell _1\le k_1}(v_{\sigma (\ell _1)}^{(1)})^{k_1-\ell _1}\prod _{1\le \ell _2\le k_2}(v_{\pi (\ell _2)}^{(2)})^{k_2-\ell _2}\) over permutations \(\sigma \in S_{k_1}\) and \(\pi \in S_{k_2}\), here \(S_k\) denotes the permutation group of \(\{1,2,\ldots ,k\}\). If \(k_1\) is large enough (the case when \(k_2\) is large is similar), for example if \(k_1>N\), the integrand is analytic for \(v_{\sigma (1)}^{(1)}\) at 0 by checking the degrees. So when we integrate \(v_{\sigma (1)}^{(1)}\), the only possible nontrivial contribution is from the residues \(v_{\sigma (1)}^{(1)}=v_{j}^{(2)}\) if \(v_j^{(2)}\) lies inside the contour of \(v_{\sigma (1)}^{(1)}\) due to the Cauchy-type factor. However, if we further integrate \(v_{j}^{(2)}\) we find each residue contribution is also zero by checking the degree of \(v_j^{(2)}\) which is \(k_1-1-n-(N-n+1)=k_1-N>0\). We remark that the proof does not rely on the explicit formula of H or the variable \({\mathrm {z}}\), and it is similar to the argument for the two-point distribution formula of TASEP (see Remark 2.8 of [31]) where they do not have the factor H.
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Acknowledgements
We would like to thank Jinho Baik, Ivan Corwin, Duncan Dauvergne, Patrik Ferrari, Kurt Johansson, Daniel Remenik and Bálint Virág for the comments and suggestions. The work was supported by the University of Kansas Start Up Grant, the University of Kansas New Faculty General Research Fund, Simons Collaboration Grant No. 637861, and NSF grant DMS-1953687.
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Liu, Z. One-point distribution of the geodesic in directed last passage percolation. Probab. Theory Relat. Fields 184, 425–491 (2022). https://doi.org/10.1007/s00440-022-01123-2
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DOI: https://doi.org/10.1007/s00440-022-01123-2
Mathematics Subject Classification
- 60K35
- 82C22