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Register a new userUniversal Formula for Extreme First Passage Statistics of Diffusion
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Sean D Lawley
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The timescales of many physical, chemical, and biological processes are determined by first passage times (FPTs) of diffusion. The overwhelming majority of FPT research studies the time it takes a single diffusive searcher to find a target. However, the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers. This fastest FPT depends on extremely rare events and has a drastically faster timescale than the FPT of a given single searcher. In this work, we prove a simple explicit formula for every moment of the fastest FPT. The formula is remarkably universal, as it holds for $d$-dimensional diffusion processes (i) with general space-dependent diffusivities and force fields, (ii) on Riemannian manifolds, (iii) in the presence of reflecting obstacles, and (iv) with partially absorbing targets. Our results rigorously confirm, generalize, correct, and unify various conjectures and heuristics about the fastest FPT.
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Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.
The time it takes the fastest searcher out of $Ngg1$ searchers to find a target determines the timescale of many physical, chemical, and biological processes. This time is called an extreme first passage time (FPT) and is typically much faster than the FPT of a single searcher. Extreme FPTs of diffusion have been studied for decades, but little is known for other types of stochastic processes. In this paper, we study the distribution of extreme FPTs of piecewise deterministic Markov processes (PDMPs). PDMPs are a broad class of stochastic processes that evolve deterministically between random events. Using classical extreme value theory, we prove general theorems which yield the distribution and moments of extreme FPTs in the limit of many searchers based on the short time distribution of the FPT of a single searcher. We then apply these theorems to some canonical PDMPs, including run and tumble searchers in one, two, and three space dimensions. We discuss our results in the context of some biological systems and show how our approach accounts for an unphysical property of diffusion which can be problematic for extreme statistics.
Consider first passage percolation with identical and independent weight distributions and first passage time ${rm T}$. In this paper, we study the upper tail large deviations $mathbb{P}({rm T}(0,nx)>n(mu+xi))$, for $xi>0$ and $x eq 0$ with a time constant $mu$ and a dimension $d$, for weights that satisfy a tail assumption $ beta_1exp{(-alpha t^r)}leq mathbb P(tau_e>t)leq beta_2exp{(-alpha t^r)}.$ When $rleq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $exp{(-(2dxi +o(1))n)}$. When $1< rleq d$, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. For $r<d$, we show that the large deviation event ${rm T}(0,nx)>n(mu+xi)$ is described by a localization of high weights around the origin. The picture changes for $rgeq d$ where the configuration is not anymore localized.
We study the distribution of first-passage functionals ${cal A}= int_0^{t_f} x^n(t), dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=text{Prob}.(mathcal{A}=A)$. This probability density has an essential singular tail as $Ato 0$ and a power-law tail $sim A^{-(n+3)/(n+2)}$ as $Ato infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $Ato 0$ it predicts the same essential singular tail as in the driftless case. For $Ato infty$ it predicts a stretched exponential tail $-ln P_n(A|x_0)sim A^{1/(n+1)}$ for all $n>0$. In the limit of large Peclet number $text{Pe}= mu x_0/(2D)gg 1$, where $mu$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-ln P_n(A|x_0)simeqtext{Pe}, Phi_nleft(z= A/bar{A}right)$, where $bar{A}=x_0^{n+1}/{mu(n+1)}$ is the mean value of $mathcal{A}$. We compute the rate function $Phi_n(z)$ analytically for all $n>-1$. For $n>0$ $Phi_n(z)$ is analytic for all $z$, but for $-1<n<0$ it is non-analytic at $z=1$, implying a dynamical phase transition. The order of this transition is $2$ for $-1/2<n<0$, while for $-1<n<-1/2$ the order of transition changes continuously with $n$. Finally, we apply the OFM to the case of $mu<0$ (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of $mathcal{A}$ coincides with the distribution of $mathcal{A}$ for $mu>0$ with the same $|mu|$.
We consider a simplified model of the continuous double auction where prices are integers varying from $1$ to $N$ with limit orders and market orders, but quantity per order limited to a single share. For this model, the order process is equivalent to two $M/M/1$ queues. We study the behaviour of the auction in the low-traffic limit where limit orders are immediately transformed into market orders. In this limit, the distribution of prices can be computed exactly and gives a reasonable approximation of the price distribution when the ratio between the rate of order arrivals and the rate of order executions is below $1/2$. This is further confirmed by the analysis of the first passage time in $1$ or $N$.
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