Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Abstract

We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of $2+1$-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari–Spohn diffusions associated with limiting Sturm–Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.

Appendices

Appendix 1: Strong approximation techniques

In order to prove (I.1), we are going to apply strong approximation techniques from [6]. By rescaling, it is sufficient to consider the case $t=1$.

In the sequel, $\hat{\mathbf {P}}^{\underline{r}}_+$ denotes the restriction of the law of the n-dimensional Brownian motion $\underline{B}$ started at $\underline{r}$ to the set $\mathbb {A}^+_{n}$.

Define

$$\begin{aligned} O_\epsilon (\underline{z}) = \left\{ \underline{y}\,:\, |y_i-z_i| \le { \frac{\epsilon }{3} } \text { for all } i\right\} . \end{aligned}$$

It follows easily from [6, Lemma 17] that

uniformly in $\underline{r},\underline{z}\in \mathbb {A}_{n, \lambda }^{+, {\mathsf r}}$. This implies that there exists a constant $c(\epsilon ,\eta ,\gamma )>0$ such that

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_{+,\lambda } (\underline{x}^\lambda (1-\gamma ) \in O_\epsilon (\underline{z}) , \max _{t\le 1-\gamma } x^\lambda _n(t) \le 2\eta ) \ge c(\epsilon ,\eta ,\gamma ) , \end{aligned}$$

(7.31)

for all $\underline{r},\underline{z}\in \mathbb {A}_{n, \lambda }^{+, {\mathsf r}}$. Since $O_\epsilon (\underline{z})$ is separated from the boundary of $\mathbb {A}_n^+$, we may choose $\gamma$ so small that the probability that the random walk $x^\lambda$ started at $\underline{y}\in O_\epsilon (\underline{z})$ has, at time $\gamma$, the value $\underline{z}$ and leaves $\mathbb {A}_{n,\lambda }^+$ before time $\gamma$ is quite small. This heuristic is made precise in [6, Lemma 29]. In our notation, we can state that result as follows: There exist $a>0$ and $c_1<\infty$ such that

$$\begin{aligned}&\hat{\mathbf {P}}^{\underline{y}}_{+,\lambda } ( \underline{x}^\lambda (\gamma ) = \underline{z}, \max _{t\le \gamma } x^\lambda _n(t) \le 2\eta ) \\&\quad \ge \hat{\mathbf {P}}^{\underline{y}}_\lambda ( \underline{x}^\lambda (\gamma ) = \underline{z}, \max _{t\le \gamma } x^\lambda _n(t) \le 2\eta ) - c_1 \gamma ^{-n/2} e^{-a\epsilon ^2/\gamma }h_\lambda ^{n} . \end{aligned}$$

By a similar argument, one can show that

$$\begin{aligned}&\hat{\mathbf {P}}^{\underline{y}}_\lambda ( \underline{x}^\lambda (\gamma ) = \underline{z}, \max _{t\le \gamma } x^\lambda _n(t) \le 2\eta ) \\&\quad \ge \hat{\mathbf {P}}^{\underline{y}}_\lambda ( \underline{x}^\lambda (\gamma ) = \underline{z}) - c_2 \gamma ^{-n/2} e^{-a\eta ^2/\gamma } h_\lambda ^{n} . \end{aligned}$$

Finally, by the standard local limit theorem,

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{y}}_\lambda ( \underline{x}^\lambda (\gamma ) = \underline{z}) \ge c_3 \gamma ^{-n/2} h_\lambda ^{n} . \end{aligned}$$

As a result, we have the bound

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{y}}_{+,\lambda } ( \underline{x}^\lambda (\gamma ) = \underline{z}, \max _{t\le \gamma } x^\lambda _n(t) \le 2\eta ) \ge c_4 h_\lambda ^{n} , \end{aligned}$$

uniformly in $\underline{y},\underline{z}\in \mathbb {A}_{n, \lambda }^{+, {\mathsf r}}$. Combining this bound with (7.31), we infer that

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_{1,+,\lambda } ( \underline{x}^\lambda (1) = \underline{z}, \max _{t\le 1} x_n^\lambda (t) \le 2\eta ) \ge c_5 h_\lambda ^{n} , \end{aligned}$$

(7.32)

uniformly in $\underline{r},\underline{z}\in \mathbb {A}_{n, \lambda }^{+, {\mathsf r}}$. $\square$

Appendix 2: Invariance principles for random walks in Weyl chambers

Conditional limit theorems and conditional invariance principles for random walks in different cones have been studied in [6] and [8]. All the results in these papers are proved in the case when the non-rescaled walk starts at a fixed point. In this paragraph, we give certain improvements of these results to the case when the starting point of the non-rescaled walk may grow (but we shall consider walks in Weyl chambers only).

More precisely, we shall consider the following subsets of the Euclidean space:

• chamber of type A: $\{x: x_1<x_2<\cdots <x_n\}$;

• chamber of type C: $\{x: 0<x_1<x_2<\cdots <x_n\}$;

• chamber of type D: $\{x: |x_1|<x_2<\cdots <x_n\}$.

Let $u_W$ denote the unique (up to a constant multiplier) positive harmonic function on W:

• if W is the chamber of type A, then $u_W(x) = \prod _{i<j} (x_j-x_i)$;

• if W is the chamber of type C, then $u_W(x) = \prod _k x_k \prod _{i<j} (x^2_j-x^2_i)$;

• if W is the chamber of type D, then $u_W(x)=\prod _{i<j} (x^2_j-x^2_i)$.

Proposition 4

Let W be a Weyl chamber of type A, C or D. Let $\tau$ be the first exit time from W, that is,

$$\begin{aligned} \tau = \inf \{t>0 \,:\, \underline{x}^\lambda (t)\notin W\}. \end{aligned}$$

Then, as $\underline{r}=\underline{r}_\lambda \rightarrow 0$,

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda ( \underline{x}^\lambda (1)\in \,\cdot \;|\, \tau >1 )\rightarrow \mu \quad \text {weakly}, \end{aligned}$$

where $\mu$ is the probability measure on W with density proportional to $u_W(x)e^{-|x|^2/2}$.

Furthermore, under $\hat{\mathbf {P}}^{\underline{r}}_\lambda$, $\underline{x}^\lambda$ converges weakly on C[0, 1] to the Brownian meander in W started at zero.

By “Brownian meander in W”, we mean a Brownian motion conditioned on staying in W up to time one. If the starting point lies inside W, then one has a condition of positive probability. However, if the starting point lies on the boundary of W, then the probability of the condition is zero and it is not at all clear how one can construct such a process. Garbit [11] has constructed Brownian meanders started at zero for a quite large class of cones. This class includes Weyl chambers.

Proof

The main difference with [6, Theorem 3] is that we find the limit for conditional distributions without determining the asymptotic behavior of $\hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau >1)$. (Recall once again that [6, Theorem 3] is proven under the assumption $\underline{r}=h_\lambda a$ for some fixed $a\in W$.)

Fix some $\epsilon \in (0,1/2)$ and define the stopping time

$$\begin{aligned} \nu _{\lambda ,\epsilon } = \inf \{t>0 \,:\, \underline{x}^\lambda (t)\in W_{\lambda ,\epsilon }\} , \end{aligned}$$

where

$$\begin{aligned} W_{\lambda ,\epsilon } = \{x\in W \,:\, {\text {dist}}(x,\partial W)\ge H_\lambda ^{-2\epsilon }\} . \end{aligned}$$

According to [6, Lemma 14],

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau> H_\lambda ^{-2\epsilon } , \nu _{\lambda ,\epsilon } > H_\lambda ^{-2\epsilon }) \le e^{-c_1 H_\lambda ^{2\epsilon }} , \end{aligned}$$

(7.33)

uniformly in $\underline{r}$. Since we consider lattice random walks, there exists $\underline{r}_0$ such that

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau>1) \ge \hat{\mathbf {P}}^{\underline{r}_0}_\lambda (\tau >1) . \end{aligned}$$

(If W is of type A or C, then we may take $r_0=h_\lambda (1,2,\ldots ,n)$, while if W is of type D, then we may take $r_0=h_\lambda (0,1,\ldots ,n-1)$.) According to [6, Theorem 1],

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}_0}_\lambda (\tau >1) \sim C_1 h_\lambda ^p , \end{aligned}$$

where p is a positive constant depending on the type of W only. Consequently,

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau >1) \ge C_2 h_\lambda ^p , \end{aligned}$$

(7.34)

uniformly in $\underline{r}$. Combining (7.33) and (7.34), we infer that

$$\begin{aligned} \frac{ \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau> H_\lambda ^{-2\epsilon } , \nu _{\lambda ,\epsilon }> H_\lambda ^{-2\epsilon }) }{ \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau >1) } \rightarrow 0, \quad \lambda \downarrow 0 , \end{aligned}$$

(7.35)

uniformly in $\underline{r}$. Furthermore, it follows from the exponential Doob inequality that

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda \big ( \max _{t\le H_\lambda ^{-2\epsilon }} |\underline{x}^\lambda (t)-\underline{x}^\lambda (0)| > \theta _\lambda \big ) \le e^{-c_2 \theta _\lambda ^2 H_\lambda ^\epsilon } , \end{aligned}$$

where $\theta _\lambda \rightarrow 0$ sufficiently slowly. This implies that, whenever $|\underline{r}|\le \theta _\lambda$,

$$\begin{aligned} \frac{ \hat{\mathbf {P}}^{\underline{r}}_\lambda \big ( \max _{t\le H_\lambda ^{-2\epsilon }} |\underline{x}^\lambda (t)|> 2\theta _\lambda \big ) }{ \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau >1) } \rightarrow 0 . \end{aligned}$$

(7.36)

It follows now from (7.35) and (7.36) that, uniformly in $\underline{r}$,

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau>1) = (1+o(1)) \, \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau >1 , \nu _{\lambda ,\epsilon }\le H_\lambda ^{-2\epsilon } , \max _{t\le \nu _{\lambda ,\epsilon }} |\underline{x}(t)| \le 2\theta _\lambda ) \end{aligned}$$

(7.37)

and

$$\begin{aligned}&\hat{\mathbf {P}}^{\underline{r}}_\lambda (\underline{x}^\lambda (1)\in A , \tau>1) \nonumber \\&\quad = (1+o(1))\, \hat{\mathbf {P}}^{\underline{r}}_\lambda (\underline{x}^\lambda (1)\in A , \tau >1 , \nu _{\lambda ,\epsilon }\le H_\lambda ^{-2\epsilon } , \max _{t\le \nu _{\lambda ,\epsilon }} |\underline{x}(t)|\le 2\theta _\lambda )\qquad \end{aligned}$$

(7.38)

for any compact $A\subset W$.

Using the Markov property at time $\nu _{\lambda ,\epsilon }$ and applying [6, Lemma 20], we obtain from (7.37) and (7.38)

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}}_\lambda (\tau >1) = (c_3+o(1))\, h_\lambda ^p\, \hat{\mathbf {E}}^{\underline{r}}_\lambda \big [u_W(\underline{x}^\lambda (\nu _{\lambda ,\epsilon });\, \nu _{\lambda ,\epsilon }\le H_\lambda ^{-2\epsilon } , \max _{t\le \nu _{\lambda ,\epsilon }} |\underline{x}(t)|\le 2\theta _\lambda \big ] \end{aligned}$$

and

$$\begin{aligned}&\hat{\mathbf {P}}^{\underline{r}}_\lambda (\underline{x}^\lambda (1) \in A , \tau >1) = (c_4+o(1))\, h_\lambda ^p\, \int _A u_W(z) e^{-|z|^2/2} \mathrm{d}z \\&\quad \times \hat{\mathbf {E}}^{\underline{r}}_\lambda \big [ u_W(\underline{x}^\lambda (\nu _{\lambda ,\epsilon }));\, \nu _{\lambda ,\epsilon }\le H_\lambda ^{-2\epsilon } , \max _{t\le \nu _{\lambda ,\epsilon }} |\underline{x}(t)|\le 2\theta _\lambda \big ] . \end{aligned}$$

Thus, the proof of the first statement is completed.

To prove the functional convergence, it suffices to repeat the proof of [8, Theorem 1] using (7.35) and (7.36) instead of the corresponding estimates therein. $\square$

Corollary 2

Let W be the chamber of type C. If $\underline{r}=\underline{r}_\lambda \rightarrow \underline{r}^*\in \partial W$, then the sequence $\hat{\mathbf {P}}^{\underline{r}}_\lambda (\underline{x}^\lambda (1)\in A \,|\, \tau >1)$ converges weakly. The densities of limiting laws on W has are uniformly bounded. Moreover, $\underline{x}^\lambda$ converges weakly on C[0, 1] towards the Brownian meander in W started at $\underline{r}^*$.

Proof

We just split the original set of random walks into a finite number of subsets in such a way that the differences of coordinates of the starting points in every block converge to zero and the differences of coordinates from different blocks stay bounded away from zero. Then, the probability that different blocks do not intersect is bounded away from zero and, consequently, the conditioning on $\{\tau >1\}$ is equivalent to conditioning every block on staying in the corresponding chamber. (If $r^*_1>0$, then every block is a random block in a chamber of type A, while if $r^*_1=0$, then the lowest block is a random walk in a chamber of type C and all other blocks are random walks in chambers of type A.) $\square$

Proof (Proof of (I.2))

[Proof of (I.2)] Assume that there exists a sequence $\underline{r}(j)$ such that

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}(j)}_{1,\lambda } \big ( \max _{t\le 1} x^\lambda _n(t)\le 2\eta \bigm | \tau >1 \big ) \rightarrow 0 . \end{aligned}$$

Since we are looking at starting points $\underline{r}$ with $r_n\le \eta$, there exists a convergent subsequence $\underline{r}(j_k)$. Let $\underline{r}^*$ denote the limiting point. It follows immediately from the usual functional CLT that the case $\underline{r}^*\in W$ is impossible. But, if $\underline{r}^*\in \partial W$, then we may use Corollary 2 to conclude that the Brownian meander in W started at $\underline{r}^*$ leaves the set $\{\underline{x}\in W \,:\, x_n\le 2\eta \}$ with probability one. However, this would contradict [18, Theorem 3.2]. Thus,

$$\begin{aligned} \inf _{\underline{r}\,:\, r_n\le \eta } \hat{\mathbf {P}}^{\underline{r}}_{1,\lambda } \big ( \max _{t\le 1} x^\lambda _n(t)\le 2\eta \bigm | \tau>1 \big ) > 0 , \end{aligned}$$

(7.39)

which implies (I.2). $\square$

Proof (Proof of (I.3))

[Proof of (I.3)] Fix some $\epsilon >0$ and define

$$\begin{aligned} W_{\le \epsilon } = \{\underline{x}\in W \,:\, |x_{i+1}-x_i|\le \epsilon \text { for some } i\ge 0 \} . \end{aligned}$$

Assume that there exists a sequence $\underline{r}(j)$ such that

$$\begin{aligned} \hat{\mathbf {P}}^{\underline{r}(j)}_{1,\lambda } \big (\underline{x}^\lambda (1)\in W_{\le \epsilon } \,;\, \max _{t\le 1} x_n^\lambda (t)\le 2\eta \bigm | \tau >1 \big ) \ge \epsilon ^{1/2} . \end{aligned}$$

We may again assume that $\underline{r}(j)$ converges to $\underline{r}^*$ and this limiting point can not lie in W. But, if $\underline{r}^*$ is on the boundary of W, then the conditions of Corollary 2 are satisfied and the contradiction follows now from the boundedness of the density of the limiting law and the fact that $vol(W_{\le \epsilon } \cap \{\underline{x}\,:\, x_n\le 2\eta \}) \le C_3 \eta ^{n-1}\epsilon$.

As a consequence we have that, for all $\epsilon$ small enough,

$$\begin{aligned} \inf _{\underline{r}\,:\, r_n\le \eta } \hat{\mathbf {P}}^{\underline{r}}_{1,+,\lambda } \big (\underline{x}^\lambda (1)\in \mathbb {A}_{n, \lambda }^{+, {\mathsf r}}, \max _{t\le 1} x^\lambda _n(t)\le 2\eta \bigm | \tau >1 \big ) \ge 1- \epsilon ^{1/2} . \end{aligned}$$

(7.40)

Combining (7.39) and (7.40), we conclude that (I.3) holds for $t=1$ and all $\epsilon$ sufficiently small. Using Brownian scaling, we conclude that (I.3) is valid for all $t>0$. $\square$