Consider a realization of a Poisson process in R^2 with intensity 1 and take a maximal up/right path from the origin to (N,N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N^chi, where chi=1/3 by a result of Baik-Deift-Johansson. Here we show that typical deviations of a maximal path from the diagonal x=y is of order N^xi with xi=2/3. This is consistent with the scaling identity chi=2xi-1, which is believed to hold in many random growth models.
The online increasing subsequence problem is a stochastic optimisation task with the objective to maximise the expected length of subsequence chosen from a random series by means of a nonanticipating decision strategy. We study the structure of optimal and near-optimal subsequences in a standardised planar Poisson framework. Following a long-standing suggestion by Bruss and Delbaen (Stoch. Proc. Appl. 114, 2004), we prove a joint functional limit theorem for the transversal fluctuations about the diagonal of the running maximum and the length processes. The limit is identified explicitly with …
We consider the exactly solvable model of exponential directed last passage percolation on $\mathbb{Z}^2$ in the large deviation regime. Conditional on the upper tail large deviation event $\mathcal{U}_δ:=\{T_{n}\geq (4+δ)n\}$ where $T_{n}$ denotes the last passage time from $(1,1)$ to $(n,n)$, we study the geometry of the polymer/geodesic $Γ_{n}$, i.e., the optimal path attaining $T_{n}$. We show that conditioning on $\mathcal{U}_δ$ changes the transversal fluctuation exponent from the characteristic $2/3$ of the KPZ universality class to $1/2$, i.e., conditionally, the smallest strip around …
We introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$. In this model, the action of a path depends on the geometry of the path and the travel time. We prove that the transversal fluctuation exponent for point-to-line action minimizers is at least $3/5$.
We investigate the asymptotic standard deviation of the Longest Common Subsequence (LCS) of two independent i.i.d. sequences of length n. The first sequence is drawn from a three letter alphabet {0,1,a}, whilst the second sequence is binary. The main result of this article is that in this asymmetric case, the standard deviation of the length of the LCS is of order square root of n. This confirms Waterman's conjecture for this special case. Our result seems to indicate that in many other situations the order of the standard deviation is also square root of n.
The problem of the fluctuation of the Longest Common Subsequence (LCS) of two i.i.d. sequences of length $n>0$ has been open for decades. There exist contradicting conjectures on the topic. Chvatal and Sankoff conjectured in 1975 that asymptotically the order should be $n^{2/3}$, while Waterman conjectured in 1994 that asymptotically the order should be $n$. A contiguous substring consisting only of one type of symbol is called a block. In the present work, we determine the order of the fluctuation of the LCS for a special model of sequences consisting of i.i.d. blocks whose lengths are unifor…
We consider the ASEP and the stochastic six vertex model started with step initial data. After a long time, $T$, it is known that the one-point height function fluctuations for these systems are of order $T^{1/3}$. We prove the KPZ prediction of $T^{2/3}$ scaling in space. Namely, we prove tightness (and Brownian absolute continuity of all subsequential limits) as $T$ goes to infinity of the height function with spatial coordinate scaled by $T^{2/3}$ and fluctuations scaled by $T^{1/3}$. The starting point for proving these results is a connection discovered recently by Borodin-Bufetov-Wheeler…
We study the fluctuations, in the large deviations regime, of the longest increasing subsequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of the longest increasing subsequence of a random permutation.
Consider two independent random strings having same length and taking values uniformly in a common finite alphabet. We study the order of the variance of the length of the longest common subsequences (LCS) of these strings when long blocks, or other types of atypical substrings, are sparsely added into one of them. Under weak conditions on the derivative of the mean LCS-curve, the order of the variance of the LCS is shown to be linear in the length of the strings. We also argue that our proofs carry over to many models used by computational biologists to simulate DNA-sequences. This is the fir…
We consider the first passage percolation model on the square lattice. In this model, $\{t(e): e{an edge of}{\bf Z}^2 \}$ is an independent identically distributed family with a common distribution $F$. We denote by $T({\bf 0}, v)$ the passage time from the origin to $v$ for $v\in {\bf R}^2$ and $B(t)=\{v\in {\bf R}^d: T({\bf 0}, v)\leq t\}.$ It is well known that if $F(0) < p_c$, there exists a compact shape ${\bf B}_F\subset {\bf R}^2$ such that for all $ε>0$, $t {\bf B}_F(1-ε) \subset {B(t)} \subset t{\bf B}_F(1+ε)$, eventually with a probability 1. For each shape boundary point $u$, we den…
We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order t^{2/3}. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order t^{1/3}, and also that the transversal fluctuations of the maximal path have order t^{2/3}. We adapt and then build on a recent study of Ham…
We study the directed polymer model in dimension ${1+1}$ when the environment is heavy-tailed, with a decay exponent $α\in(0,2)$. We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature $β=β_n$ vanishes as the size of the system $n$ goes to infinity. When $α\in(1/2,2)$, we show that all possible transversal fluctuations $\sqrt{n} \leq h_n \leq n$ can be achieved by tuning properly $β_n$, allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by…
In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they…
The problem of the order of the fluctuation of the Longest Common Subsequence (LCS) of two independent sequences has been open for decades. There exist contradicting conjectures on the topic, due to Chvatal - Sankoff in 1975 and Waterman in 1994. In the present article, we consider a special model of i.i.d. sequences made out of blocks. A block is a contiguous substring consisting only of one type of symbol. Our model allows only three possible block lengths, each been equiprobable picked up. In this context, we introduce a random operation (random modification) on the blocks of one of the seq…
In discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $\mathbb Z^2$, and each finite upright path in $\mathbb Z^2$ is ascribed the weight given by the sum of values of its vertices. The weight of a collection of disjoint paths is the sum of its members' weights. The notion of a geodesic, a maximum weight path between two vertices, has a natural generalization concerning several disjoint paths: a $k$-geodesic watermelon in $[1,n]^2\cap\mathbb Z^2$ is a collection of $k$ disjoint paths contained in this square that has maximum weight among …
The problem of determining the correct order of fluctuation of the optimal alignment score of two random strings of length $n$ has been open for several decades. It is known that the biased expected effect of a random letter-change on the optimal score implies an order of fluctuation linear in $\sqrt{n}$. However, in many situations where such a biased effect is observed empirically, it has been impossible to prove analytically. The main result of this paper shows that when the rescaled-limit of the optimal alignment score increases in a certain direction, then the biased effect exists. On the…
We show that the number of renewals up to time $t$ exhibits distributional fluctuations as $t\to\infty$ if the underlying lifetimes increase at an exponential rate in a distributional sense. This provides a probabilistic explanation for the asymptotics of insertion depth in random trees generated by a bit-comparison strategy from uniform input; we also obtain a representation for the resulting family of limit laws along subsequences. Our approach can also be used to obtain rates of convergence.
In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance $L$, geodesics exhibit deviations from their mean length $L$ that are of order $L^χ$, while the transversal fluctuations, known as wandering, grow as $L^ξ$. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents $ξ=3/5$ and $χ= 1/5$, or $ξ=7/10$ and $χ= 2/5$, depending only on coarse details of the specific connectivity laws used. Also, the travel time fluctuations are Gaussian, ra…
